NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

NEAR OPTIMAL BOUNDS IN FREIMAN’S
THEOREM
TOMASZ SCHOEN
Abstract
We prove that if for a finite set A of integers we have |A + A| ≤K|A|, thenAis 1+C(log K)−1/2
contained in a generalized arithmetic progression of dimension at most K
1+C(log K)−1/2
and of size at most exp(K )|A| for some absolute constant C. We also
discuss a number of applications of this result.
1. Introduction
Freiman’s theorem [8] is one of the most fundamental theorems in additive number
theory. It asserts that if for a finite A ⊆ Z we have |A + A| ≤K|A|, thenA is
a subset of a d-dimensional generalized arithmetic progression P1 +···+Pd with
|P1|···|Pd| =|A|f , where d and f depend only on K (here Pi stands for usual
arithmetic progressions). This result provides a complete description of sets for which
the sumset is not much larger than the set itself. Freiman’s theorem has a plethora
of deep applications to many important problems, such as the one recently described
by Gowers in [9] and[11], who used it to get an effective version of Szemerédi’s
theorem (for further works that make important use of Freiman’s result, see, e.g.,
[6], [5], [25], [26]–[28]). However, most applications require a quantitative version of
Freiman’s theorem. It is easy to see that one cannot do better than d(K) ≤ K − 1
and f (K) = e O(K) . The estimates which followed from Freiman’s original proof
were quite poor. The first useful estimates for d and f were provided by Ruzsa
[19], whose ingenious argument combines techniques from different fields. Ruzsa’s
bounds were good enough for many applications and his approach paved the way
for further improvements. The next substantial progress was made by Chang [3],
who showed that one can take d ≤ K 2+o(1) and f (K) ≤ exp(K 2+o(1) ). Her proof
coupled Ruzsa’s main ideas with Chang’s so-called spectral lemma and covering
lemma. Further improvements were due to Sanders [21, Theorem 1.3], who used a
much-expanded version of Bogolyubov-Ruzsa’s lemma [20, Lemma 2.1] to obtain the
best currently known bounds d(K) ≤ K 7/4+o(1) and f (K) ≤ exp(K 7/4+o(1) ). The main
DUKE MATHEMATICAL JOURNAL
Vol. 158, No. 1, c○ 2011 DOI 10.1215/00127094-1276283
Received 16 February 2010. Revision received 13 July 2010.
2010 Mathematics Subject Classification. Primary 11P70; Secondary 11B25.
Author’s work supported in part by Ministerstwo Nauki i Szkolnictwa Wy˙zszego grant N N201 543538.
1

2 TOMASZ SCHOEN
result of this article is the following theorem, which gives nearly optimal estimates
for d and f .
THEOREM 1
There exists an absolute constant C such that each set A ⊆ Z satisfying |A +
A| ≤K|A| is contained in a generalized arithmetic progression of dimension at
1+C(log K)−1/2
most d(K) and size at most f (K)|A|, with d(K) ≤ K and f (K) ≤
1+C(log K)−1/2
exp(K ).
Our approach is yet another follow-up of Ruzsa’s original argument, and our improvement
relies on further refinement of Bogolyubov-Ruzsa’s lemma. A new addition
to this setup is an elementary result, Lemma 3, which roughly says that one can
find two sets “closely related” to A which have much better additive properties. As
a consequence of our method, we infer that there is a proper arithmetic progression
P of dimension Oε(Kε ) and size bounded from above by KO(1) |A| such that
|A ∩ P |≥exp(−Oε(Kε ))|A| (for a precise statement, see Theorem 8 in Section 4).
This fact has been already anticipated by Chang [3], Gowers [10], and Green and
Tao [14, Conjecture 1.6], and is often required in applications. Remarkably, Green
and Tao [14] proved that this result is equivalent to an inverse theorem for Gowers’s
U 3-norm. It can also be considered a first step toward resolving Freiman-Ruzsa’s
polynomial conjecture. We also note that, using Freiman-Bilu’s [1] argument (see also
[3]), one cannot deduce from Theorem 1 that A is contained in a progression with
1+C(log K)−1/2
linear dimension and size at most exp(K )|A|. The reason is that we cannot
guarantee our progression to be proper. Our proof can be modified to achieve it as
well, but in this case the size of the progression goes up to exp(K2+o(1) )|A|.
Finally, our method can be directly applied to Green-Ruzsa’s [13, Theorem 1.1]
proof of Freiman’s theorem in the “torsion” case, giving the following statement, the
proof of which we omit here (further improvement of this theorem will appear in [7]).
THEOREM 2
Let G be an arbitrary Abelian group, and let A ⊆ G satisfy |A + A| ≤K|A|. Then A
is contained in a coset progression P +H of dimension d(K) ≤ (K+2) 3+C(log(K+2))−1/2
and size f (K)|A| ≤exp((K + 2) 3+C(log(K+2))−1/2
)|A| for some absolute constant C.
This paper is organized as follows. Section 2 contains some basic definitions and facts
that we use here. The main part of the paper is Section 3, where we prove Lemma 3,
which is key to our argument. In order to keep the paper self-contained, we briefly
outline the rest of Ruzsa’s argument (with Chang’s refinements) in Section 4. We
conclude with some applications of our results and a few comments on our approach
and its possible further developments.

NEAR OPTIMAL BOUNDS IN FREIMAN’S THEOREM 3
2. Preliminaries
This part of the article contains basic definitions and notation we will use later on.
As usual, we set
A + B ={a + b : a ∈ A, b ∈ B},
and the k-fold sumset of A is denoted by kA.Byageneralized arithmetic progression
of dimension d, we mean every set of the form P = P1+···+Pd, where P1,...,Pd are
usual arithmetic progressions. The size of P is defined as the product |P1|···|Pd|.The
dimension and size of progression P are denoted by dim(P ) and size(P ), respectively.
If each x ∈ P has unique representation x = p1 +···+pd, pi ∈ Pi, thenwesay
that P is proper. Then the cardinality of P is equal to its size.
Let G, H be Abelian groups, and let A ⊆ G, B ⊆ H . We say that A is Fkisomorphic
(i.e., Freiman isomorphic of order k) toB if there exists a bijective map
ϕ : A → B such that
if and only if
x1 +···+xk = y1 +···+yk
ϕ(x1) +···+ϕ(xk) = ϕ(y1) +···+ϕ(yk)
for every x1,...,xk,y1,...,yk ∈ A.
We call (x,x ′ ,y,y ′ ) an additive quadruple if x + y = x ′ + y ′ . The number of
additive quadruples in X2 × Y 2 is denoted be E(X, Y ).
In this paper, by Zn we always mean Z/nZ. The Fourier coefficients of the
indicator function of a set X ⊆ Zn are defined by
ˆX(s) =
e −2πixs/n ,
x∈X
where s ∈ Zn. Parseval’s formula states that n−1
s=0 | ˆX(s)| 2 =|X|n. We observe also
that, for X, Y ⊆ Zn, wehave
E(X, Y ) = 1 n−1
| ˆX(s)|
n
2 | ˆY (s)| 2 . (1)
s=0
Our argument makes use of Plünnecke-Ruzsa’s inequality, which says, in particular
that if |A + A| ≤K|A|, then for all k, l ∈ N, wehave
|kA − lA| ≤K k+l |A|.

4 TOMASZ SCHOEN
By α, we denote the distance from α to the nearest integer. For Ɣ ⊆ Zn and
γ>0, the Bohr set B(Ɣ, γ ) is defined as
B(Ɣ, γ ) ={x ∈ Zn : gx/n ≤γ for every g ∈ Ɣ}.
3. The main lemma
The key idea of our approach is to identify a relatively large subset of A that has
better additive properties than A. One possible way would be to find A ′ ⊆ A such that
|kA ′ |≤K εk |A ′ | for not too-big k. A result of this kind was proved in [17, Lemma
1], but under a stronger assumption on A. Unfortunately, this method does not seem
to work in our general setting. Let us remark, however, that Bogolyubov-Ruzsa’s
lemma uses the number of additive quadruples rather than the size of the subset.
This suggests the following approach: using the assumption |A + A| ≤K|A|, one
should find for some small k a relatively dense set X, Y ⊆ kA − kA with E(X, Y )
much bigger than one can expect from the size of A + A. As far as we know, the
first such result was proved by Katz and Koester [15]. They showed that there is
B ⊆ A ± A, |B| ≫|A|/K C with E(B,B) ≫|B| 3 /K 36/37 . Lemma 3 provides a
much better estimate for E(X, Y ) in a “highly nonsymmetric” case, when one set is
large and the other is small (a similar result for a diagonal case would give a much
better result—see our concluding remarks below). Finally, let us also mention that
Sanders in [22], and [23] (seealso[24] for related theorems), used a similar result
for a different purpose—that of improving the bounds in many summed versions of
Roth-Meshulam’s theorem.
LEMMA 3
Suppose that A is a subset of an Abelian group and that |A+A| ≤K|A|. Then for every
ε>0, there are sets X ⊆ A and Y ⊆ A + A such that |X| ≥(2K2 ) −21/ε|A|,
|Y |≥
|A|, and E(X, Y ) ≥ K−2ε |X| 2 |Y |.
Proof
For a set B ⊆ A, denote by D = D(B) the set of all t ∈ B − B which has at least
|B| 2 |B|2 |B|2
≥ ≥
2|B − B| 2|A − A| 2K2 |A|
representations as b − b ′ , b,b ′ ∈ B (the last inequality follows from Plünnecke-
Ruzsa’s inequality). Note that |D| ≥|B|/2.
We first show that there exists a set B ⊆ A such that |B| ≥(2K 2 ) −21/ε +1 |A| and
|A + Bt| ≥K −ε |A + B| (2)
for all t ∈ D = D(B), where Bt = B ∩ (B + t). We construct B in the following
iterative procedure. We start with A 1 = A. Now suppose that A l with

NEAR OPTIMAL BOUNDS IN FREIMAN’S THEOREM 5
|Al |≥(2K2 ) −2l−1 +1 |A| has already been defined. If there exists t0 ∈ D(Al ) with
|A + Al t0 |

6 TOMASZ SCHOEN
Since |D ′ |≥|B|/2 ≥ (2K2 ) −21/ε|A|,
the assertion follows for X = D ′ and Y =
A + B. ✷
Proof of Theorem 1
In order to show Theorem 1, one needs to mimic Ruzsa-Chang’s proof using Lemma 3.
In order to make the article self-contained, we briefly sketch their argument.
Let n be a prime satisfying
48|12A − 12A| >n>24|12A − 12A|.
By Plünnecke-Ruzsa’s inequality, it follows that n ≤ 48K 24 |A|. By Ruzsa’s lemma
(see [18, Theorem 2]), there exists a set A ′ ⊆ A such that |A ′ |≥|A|/12, whichis
F12-isomorphic to T ⊆ Zn. Clearly, |T + T |≤12K|T |. Let X ⊆ T and Y ⊆ T + T
be sets such that the assertion of Lemma 3 holds with
Thus, we have
ε = (log K) −1/2 .
|X| ≥ 2(12K) 2 −2 1/ε
|T |≥2 −10 (288K 2 ) −21/ε
K −24 n.
The next ingredient of our approach is Chang’s spectral lemma (see [3]and[13]).
For Ɣ ⊆ Zn, define
Span(Ɣ) = εgg : εg ∈{−1, 0, 1} .
g∈Ɣ
Then the spectral lemma can be stated as follows.
LEMMA 4([3, Lemma 3.1])
Let X ⊆ Zn, and let ={r ∈ Zn : | ˆX(r)| ≥λ|X|}. Then there is a set Ɣ ⊆ Zn such
that |Ɣ| ≤2λ −2 log(n/|X|) and ⊆ Span(Ɣ).
Let us emphasize at this point that Chang’s spectral lemma is absolutely crucial for
our argument. While in Chang’s paper one can omit it and still get reasonable bounds
d(K) ≤ K O(1) and f (K) ≤ exp(K o(1) ), without Lemma 4 our approach completely
fails. In fact, as it is easy to see, Chang’s spectral lemma is especially useful if λ is
much bigger than the density of X, and we apply it precisely in such a case using its
full potential.
Our goal is to get a version of Bogolyubov-Ruzsa’s lemma for sets X and Y .

NEAR OPTIMAL BOUNDS IN FREIMAN’S THEOREM 7
LEMMA 5
The set X + Y − X − Y contains a Bohr set B(Ɣ, γ ) such that |Ɣ| ≪K 3ε log K and
γ ≫ K −3ε log −1 K.
Proof
Define
= r ∈ Zn : | ˆX(r)| ≥K −ε |X|/2 .
We will show that B = B(, 1/6) ⊆ X + Y − X − Y .Letr(x) be the number of
representations of x in X + Y − X − Y .Ifx ∈ B, then, by (1) and Parseval’s formula,
we have
r(x) = 1 n−1
| ˆX(s)|
n
2 | ˆY (s)| 2 e 2πixs/n
≥ 1
n
≥ 1
2n
s=0
s∈
s∈
s=0
| ˆX(s)| 2 | ˆY (s)| 2 cos(2πxs/n) − 1
n
| ˆX(s)| 2 | ˆY (s)| 2 − 1
n
| ˆX(s)| 2 | ˆY (s)| 2
s∈
| ˆX(s)| 2 | ˆY (s)| 2
s∈
≥ 1 n−1
| ˆX(s)|
2n
2 | ˆY (s)| 2 − 3
| ˆX(s)|
2n
2 | ˆY (s)| 2
s∈
> E(X, Y )/2 − 3
8n K−2ε |X| 2
n−1
| ˆY (s)| 2
s=0
≥ E(X, Y )/2 − (3/8)K −2ε |X| 2 |Y | > 0.
By Chang’s spectral lemma (Lemma 4), there is a set Ɣ such that |Ɣ| ≪
K 2ε log(n/|X|) ≪ K 3ε log K and ⊆ Span(Ɣ), so that
B Ɣ, 1/(6|Ɣ|) ⊆ B(, 1/6) ⊆ X + Y − X − Y. ✷
Remark. We proved Bogolyubov-Ruzsa’s lemma in a standard way, but one can easily
strengthen the assertion by removing one summand. Indeed, by (3), we have
1 n−1
ˆX(s)| ˆY (s)|
n
s=0
2 e −2πib0s/n −ε
≥ K |X||Y |.
Therefore, using a similar argument as in the proof of Lemma 5, we infer that a shift
of B(Ɣ, γ ) is contained in X + Y − Y.

8 TOMASZ SCHOEN
Let us first recall Ruzsa’s lemma on the geometry of numbers (see [20, Lemma
3.2]).
LEMMA 6
Let B(Ɣ, γ ) be a Bohr set in Zn. Then there is a proper progression P ⊆ B(Ɣ, γ ) of
dimension |Ɣ| and size at least (γ/|Ɣ|) |Ɣ| n.
Thus by Lemmas 5 and 6, one can find a proper progression P ⊆ X + Y − X − Y ⊆
3T − 3T with dim(P ) ≤|Ɣ| ≪K 3ε log K and
size(P ) ≥ exp −K 3ε (log K) 3/2+o(1) n.
Let P ′ betheimageofP under a considered F12-isomorphism between A ′ and T .It
induces an F2-isomorphism between P and P ′ , which implies that P ′ is also proper.
For the last step of the argument we need Chang’s covering lemma [3,Step4].
LEMMA 7
If |A + A| ≤K|A| and |B + A| ≤L|B|, then there are sets S1,...,Sk each of size
≤ 2K such that A ⊆ B − B + (S1 − S1) +···+(Sk − Sk) and k ≤ 1 + log(KL).
Now, we have
so that
|P ′ + A| ≤|5A − 4A| ≤K 9 |A| ≤K 9 exp K 3ε (log K) 3/2+o(1) |P ′ |
A ⊆ P ′ − P ′ + (S1 − S1) +···+(Sk − Sk)
with k ≤ K 3ε (log K) 5/2+o(1) . Finally, A is contained in a progression Q with
and
1+C(log K)−1/2
dim(Q) ≤ dim(P ) + 2kK ≤ K
size(Q) ≤ 2 dim(P ′ 1+C(log K)−1/2
) ′ 2kK K
size(P )3 ≤ e |A|
for some constant C. This completes the proof of Theorem 1.
4. Some applications
We focus on the consequences of a new version of Bogolyubov-Ruzsa’s lemma in
Freiman’s theorem as one of the most important applications. However, as an immediate
consequence of the results obtained in Section 3, we obtain the following step in
the direction of Freiman-Ruzsa’s polynomial conjecture, which may also have other
applications.

NEAR OPTIMAL BOUNDS IN FREIMAN’S THEOREM 9
THEOREM 8
There exists an absolute constant C such that, if A ⊆ Z satisfies |A+A| ≤K|A|, then
C(log K)−1/2
there is a proper generalized arithmetic progression P with dim(P ) ≤ K
and size(P ) ≤ 48K24 C(log K)−1/2
|A| such that |A ∩ P |≥exp(K )|A|.
Using Theorem 8 and Theorem 1, one can directly improve many results based on
Freiman’s theorem; we mention only some of them here. The most exciting case is
Gowers’s theorem; however, here the effect is not so impressive. It is proved in [11,
Theorem 18.6] that, for every k ∈ N, the density of a subset of {1,...,n} which does
not contain any k-term arithmetic progression does not exceed
(log log n) −ck ,
where ck = 2−2k+9. Applying our version of Bogolyubov-Ruzsa’s lemma, one can at
most increase the value of ck. On the other hand, one can deduce from Lemma 5 (and
in view of the Remark) that, for each set A ⊆{1,...,n} of density δ =|A|/n, one
can find an arithmetic progression of length
1/δ)1/2
e−c(log
cδn
in 5A for some absolute constant c>0, beating the previous best bound cδn cδ1/3
obtained by Sanders [20]. Another interesting consequence is a result related to a
problem considered by Konyagin and Łaba [16, Corollary 3.7]. Inserting our estimates
in Sanders’s proof (see [21, Theorem 1.2]), we infer that there exists c>0 such that,
for every finite set A ⊆ R and every transcendental α ∈ R, wehave
|A + α · A| ≥(log |A|) c log log |A| |A|.
Let us also formulate our version of Bogolyubov-Ruzsa’s lemma for F n 2
follows.
THEOREM 9
There are absolute constants C1,C2 such that, if A ⊆ Fn 2
to read as
has density δ, then6A
√
contains a subspace H of codimension C1 exp(C2 log(1/δ)).
We refer the reader to [12] for other consequences of Lemma 3 in F n 2 . The last
application provides progress toward Ruzsa’s conjecture, which states that, for every
finite set of squares A, wehave|A + A| ≫|A| 2−ε . Currently, the best estimate
|A + A| ≥(log |A|) 1/12 |A| is due to Chang [4].

10 TOMASZ SCHOEN
THEOREM 10
There is a positive constant c such that, for every finite set A ⊆{1 2 , 2 2 ,...}, we have
|A + A| ≥(log |A|) c log log |A| |A|.
Proof
Set |A + A| =K|A|. By Theorem 8, there is a proper generalized arithmetic pro-
C(log K)−1/2
gression P = P1 +···+Pd with d ≤ K and size(P ) ≤ 48K24 |A| such
that
K)−1/2
−KC(log
|A ∩ P |≥e |A|.
Suppose that |P1| =maxi |Pi| ≥|P | 1/d , and write P =
x∈P2+···+Pd (P1 + x). Now,
we make use of a theorem of Bombieri and Zannier [2, Theorem 1] stating that an
arithmetic progression of length k contains O(k2/3+ε ) squares. Therefore, we have
|A ∩ P |≤
|(P1 + x) ∩ A| ≪
|P1| 3/4 ≤ (48K 24 |A|) 1−1/(4d) ,
hence
x∈P2+···+Pd
x∈P2+···+Pd
K ≥ (log |A|) c log log |A| ,
and the assertion follows. ✷
It would be interesting to strengthen Lemma 3 since, clearly, each such improvement
would result in better bounds in Freiman’s theorem. The argument presented here has
one major weakness—it works only for small sets X (one would prefer to have an
analog of this result which is also valid for sets X of roughly the same order as Y ).
Indeed, if one could prove a version of Lemma 3 with |X| ∼|Y | and, say ε = 1/20,
then Balog-Szemerédi-Gowers’s theorem would give |X ′ +X ′ |≤K 1−c |X ′ |, for some
X ′ ⊆ X and c>0. Then, by iterating this process, one would most probably get a
result very close to Freiman-Ruzsa’s polynomial conjecture.
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Faculty of Mathematics and Computer Science, Adam Mickiewicz University, 61-614 Poznań,
Poland; schoen@amu.edu.pl

SUR LA NON-DENSITÉ DES POINTS ENTIERS
PASCAL AUTISSIER
Résumé
On donne des résultats de non-densité pour les points entiers sur des variétés affines,
dans l’esprit de la conjecture de Lang-Vojta. En particulier, soit X une variété projective
de dimension d ≥ 2 sur un corps de nombres K (resp., sur C). Soit H la somme de
2d diviseurs amples sur X qui se coupent proprement. On montre que tout ensemble
de points quasi-entiers (resp., toute courbe entière) sur X − H est non Zariski-dense.
Abstract
We give nondensity results for integral points on affine varieties, in the spirit of the
Lang-Vojta conjecture. In particular, let X be a projective variety of dimension d ≥ 2
over a number field K (resp., over C). Let H be the sum of 2d properly intersecting
ample divisors on X. We show that any set of quasi-integral points (resp., any integral
curve) on X − H is not Zariski-dense.
1. Introduction
On s’intéresse ici aux solutions à coordonnées (quasi-)entières de systèmes d’équations
polynomiales à coefficients dans un corps de nombres K: on prouve qu’une variété
projective sur K privée de suffisamment d’hypersurfaces n’a pas beaucoup de points
(quasi-)entiers. On a également les résultats analogues en géométrie hyperbolique:
une telle variété (sur C) n’a pas de courbe entière non dégénérée.
Plus précisément, soit S un ensemble fini de places de K. On note OKS l’anneau
des S-entiers de K. Cet article s’inscrit dans le cadre d’un problème important de la
géométrie arithmétique.
CONJECTURE 1.1 ([6, Conjecture 4.2], [13, Conjecture 3.4.3])
Soit X une variété projective lisse sur K de diviseur canonique KX.SoitD un diviseur
effectif sur X, à croisements normaux. Posons Y = X − D. On suppose KX + D big
(par exemple, ample) sur X. Alors tout ensemble E ⊂ Y (K) S-entier sur Y est non
Zariski-dense dans Y .
DUKE MATHEMATICAL JOURNAL
Vol. 158, No. 1, c○ 2011 DOI 10.1215/00127094-1276292
Received 14 December 2009. Revision received 16 July 2010.
2010 Mathematics Subject Classification. Primary 14G25; Secondary 11J97, 11G35.
13

14 PASCAL AUTISSIER
L’hypothèse sur KX + D est une condition de positivité de ce diviseur. Notons que
cette conjecture est encore largement ouverte: le cas où X = P2 K n’est par exemple pas
connu. Levin en a proposé des cas particuliers intéressants, lorsque D a suffisamment
de composantes irréductibles.
CONJECTURE 1.2 ([8, Conjecture 5.3A])
Soit X une variété projective sur K de dimension d ≥ 1. Soient D1,...,Dr des
diviseurs effectifs big sur X qui se coupent proprement. Posons Y = X−D1∪···∪Dr.
On suppose r ≥ d + 2. Alors aucun ensemble S-entier sur Y n’est Zariski-dense
dans Y .
Pour une explication du lien entre 1.1 et 1.2, on pourra consulter [8, paragraphe 14.2].
Un théorème classique de Siegel montre cette conjecture lorsque X est une courbe.
Le cas des surfaces lisses est obtenu par Levin [8, théorème 6.2A.c].
Par ailleurs, c’est une conséquence directe du théorème du sous-espace (voir
Schmidt [11], Schlickewei [10]) lorsque X = P d K et les Di sont des hyperplans
en position générale. Plus généralement, cet énoncé est connu de Vojta [15, corollaire
0.3] pour X lisse et r ≥ d + ρ + 1, oùρ désigne le nombre de Picard de
X K.
Dans un article récent, Corvaja, Levin, et Zannier [2] démontrent la conjecture
1.2 avec d 2 − d + 1 diviseurs Di amples sur X lisse (d ≥ 3). On se propose ici
d’améliorer ce résultat en considérant un nombre de diviseurs linéaire en d (au lieu
de quadratique).
THÉORÈME 1.3
Soit X une variété projective sur K de dimension d ≥ 2. Soient D1,...,D2d des
diviseurs effectifs amples sur X qui se coupent proprement. Posons Y = X − D1 ∪
···∪D2d. Alors tout ensemble E ⊂ Y (K) S-entier sur Y est non Zariski-dense dans
Y .
On l’obtient comme corollaire d’un critère (théorème 2.11) qui donne des conditions
géométriques de non-Zariski-densité des points S-entiers.
La démonstration s’inspire de la méthode introduite par Corvaja et Zannier dans
le cas des courbes (voir [3]) et des surfaces (voir [4]). Cette méthode, dont l’ingrédient
arithmétique principal est le théorème du sous-espace, a ensuite été étendue en dimension
supérieure par Levin [8], l’auteur [1], et par Corvaja, Levin, et Zannier [2].
L’idée nouvelle ici est de considérer un faisceau cohérent codant les ordres
d’annulation optimaux en les Di et de prouver un théorème de convexité (théorème
3.6) sur les sections globales de ce faisceau.
À titre d’application du critère 2.11, on montre en outre l’énoncé suivant.

SUR LA NON-DENSITÉ DES POINTS ENTIERS 15
THÉORÈME 1.4
Soit X une variété projective sur K de dimension d ≥ 2. Soient D1,...,Dd+2 des
diviseurs effectifs non nuls sur X qui se coupent proprement. Posons L = d+2 i=1 Di
et Y = X − D1 ∪···∪Dd+2. On suppose que L − (d + 1)Di est ample pour tout
i ∈{1,...,d+ 2}. Alors aucun ensemble S-entier sur Y n’est Zariski-dense dans Y .
L’hypothèse sur les L − (d + 1)Di est vérifiée lorsque les Di vivent dans un cône
“suffisamment étroit” du groupe de Néron-Severi de X.
Par ailleurs, Vojta [13] adéveloppé un “dictionnaire” entre la géométrie diophantienne
et la théorie de Nevanlinna: l’étude des points S-entiers sur les variétés sur
K est mise en analogie avec l’étude des courbes entières sur les variétés complexes.
À titre d’exemple, le théorème de Siegel sur les courbes est l’analogue du théorème
de Picard.
Pour l’étayer, on obtient aussi le critère (théorème 2.12) qui, dans ce dictionnaire,
correspond au théorème 2.11.
La section 2 décrit les critères de quasi-hyperbolicité, qui sont démontrés à la
section 4. La section 3 donne le théorème de convexité 3.6 (ainsi qu’un lemme
d’algèbre locale prouvé à la section 6). On en déduit les théorèmes 1.3 et 1.4 à la
section 5.
2. Définitions et énoncés
Soit K un corps. Commençons par quelques définitions de géométrie.
Conventions
On appelle variété sur K tout schéma quasi-projectif et géométriquement intègre sur
K. Le mot “diviseur” sous-entend “diviseur de Cartier”.
Soit X une variété projective sur K de dimension d ≥ 1. Lorsque L est un
diviseur sur X tel que h 0 (X, L) ≥ 1, ondésigne par BL le lieu de base de Ɣ(X, L) et
par L : X − BL → P(Ɣ(X, L)) le morphisme défini par Ɣ(X, L). Pour tout diviseur
effectif D sur X, on note 1D la section globale de OX(D) qu’il définit.
Définition 2.1
Un diviseur L sur X est dit grand lorsque h 0 (X, L) ≥ 1 et L est génériquement fini.
Définition 2.2
Soient D1,...,Dr des diviseurs effectifs sur X. On dit que D1,...,Dr se coupent
proprement lorsque pour toute partie I non vide de {1,...,r}, la section globale
(1Di)i∈I de
i∈I OX(Di) est régulière (autrement dit, pour tout x ∈
i∈I Di, en
notant ϕi une équation locale de Di en x, les(ϕi)i∈I forment une suite régulière de
l’anneau local OX,x).

16 PASCAL AUTISSIER
Remarque 2.3
Supposons X de Cohen-Macaulay (par exemple, lisse sur K); alors d’après [5, lemme
A.7.1], les diviseurs D1,...,Dr se coupent proprement si et seulement si pour toute
partie I non vide de {1,...,r}, le fermé
i∈I Di est purement de codimension #I
dans X (éventuellement vide).
Définition 2.4
Lorsque L est un diviseur sur X tel que q = h 0 (X, L) ≥ 1 et E un diviseur effectif
non nul sur X, on pose
α(L, E) = 1
q
h 0 (X, L − kE).
Introduisons ensuite les notions d’hyperbolicité étudiées dans cet article.
k≥1
Convention 2.5
Lorsque K est un corps de nombres et v une place de K,ondésigne par Kv le complété
de K en v, et on normalise la valeur absolue ||v de sorte que |x|v =|x| [Kv:R] si v est
archimédienne et |p|v = p −[Kv:Qp] si v est p-adique.
Lorsque K est un corps de nombres et S un ensemble fini de places de K, on note
OK,S l’anneau des S-entiers de K (i.e., l’ensemble des x ∈ K tels que |x|v ≤ 1 pour
toute place finie v/∈ S).
Soit K un corps de nombres.
Définition 2.6
Soient Y une variété sur K, K ′ une extension finie de K,etS un ensemble fini de places
de K ′ . Un ensemble E ⊂ Y (K ′ ) est dit S-entier sur Y lorsqu’il existe un OK ′ ,S-schéma
intègre et quasi-projectif Y de fibre générique YK ′ tel que E ⊂ Y(OK ′ ,S).
Définition 2.7
Soient Y une variété sur K,etK ′ une extension finie de K. Un ensemble E ⊂ Y (K ′ )
est dit quasi-entier sur Y lorsqu’il existe un ensemble fini S de places de K ′ tel que E
soit S-entier sur Y .
Définition 2.8
Soit Y une variété sur K. On dit que Y est arithmétiquement quasi-hyperbolique
lorsqu’il existe un fermé Z = Y tel que pour toute extension finie K ′ de K et tout
ensemble quasi-entier E ⊂ Y (K ′ ) sur Y , l’ensemble E − Z(K ′ ) soit fini.

SUR LA NON-DENSITÉ DES POINTS ENTIERS 17
Définition 2.9
Soit U une variété complexe. Une courbe entière sur U est une application holomorphe
f : C → U non constante.
Définition 2.10
Soit U une variété complexe. On dit que U est Brody quasi-hyperbolique lorsqu’il
existe un fermé Z = U tel que pour toute courbe entière f sur U, onaitf (C) ⊂ Z.
Terminons cette section avec les énoncés principaux de ce travail. Ces critères de
quasi-hyperbolicité généralisent (et simplifient) des résultats de Levin [8, théorèmes
8.3A et 8.3B] et de l’auteur (voir [1,théorèmes 3.3 et 3.5]).
THÉORÈME 2.11
Soit X une variété projective sur K de dimension d ≥ 1. Soient D1,...,Dr des
diviseurs effectifs non nuls sur X qui se coupent proprement. Posons Y = X − D1 ∪
···∪Dr. Soit m ≥ 1 un entier. On suppose que le diviseur L = m r i=1 Di est grand
sur X et que α(L, Di) >mpour tout i ∈{1,...,r}. AlorsYest arithmétiquement
quasi-hyperbolique.
THÉORÈME 2.12
Soit V une variété complexe projective de dimension d ≥ 1. Soient D1,...,Dr des
diviseurs effectifs non nuls sur V qui se coupent proprement. Posons U = V − D1 ∪
···∪Dr. Soit m ≥ 1 un entier. On suppose que le diviseur L = m r i=1 Di est
grand sur V et que α(L, Di) >mpour tout i ∈{1,...,r}. AlorsUest Brody
quasi-hyperbolique.
3. Préliminaires
Soient K un corps et W un K-espace vectoriel de dimension finie.
Définitions
Une filtration de W est une famille décroissante F = (Fx)x∈R+ de sous-espaces
vectoriels de W telle que Fx ={0} pour tout x assez grand. Une base B de W est
dite adaptée à la filtration F lorsque B ∩ Fx est une base de Fx pour tout réel x ≥ 0.
LEMME 3.1
Soient F et G deux filtrations de W . Il existe alors une base de W adaptée à F et à
G .
Démonstration
C’est le lemme 3.2 de [4].
Soit r un entier ≥ 1. On note ici l’ensemble des t = (t1,...,tr) ∈ Rr + tels que
t1 +···+tr = 1.

18 PASCAL AUTISSIER
Définition 3.2
Une partie N de N r est dite saturée lorsque a + b ∈ N pour tout a ∈ N r et tout
b ∈ N.
Soit A un anneau local noethérien. Soit (ϕ1,...,ϕr) une suite régulière de A. Pour
toute partie saturée N de Nr ,ondésigne par I(N) l’idéal de A engendré par l’ensemble
{ϕ b1
1 ,...,ϕbr r , b ∈ N}. On aura besoin d’un résultat d’algèbre locale (qui ne sera
démontré qu’àlasection6).
LEMME 3.3
Soient N et M deux parties saturées de N r . On a alors l’égalité d’idéaux
I(N) ∩ I(M) = I(N ∩ M).
Remarque 3.4
On utilisera dans la suite le cas particulier suivant du lemme 3.3: pour t ∈ et
x ∈ R+, notons N(t,x) l’ensemble des b ∈ Nr tels que t1b1 +···+trbr ≥ x. Ona
alors l’inclusion
I N(t,x) ∩ I N(u,y) ⊂ I N(λt + (1 − λ)u,λx+ (1 − λ)y)
pour tous t, x, u, y, et tout λ ∈ [0, 1], puisque N(t,x) ∩ N(u,y) ⊂ N(λt + (1 −
λ)u,λx+ (1 − λ)y).
Maintenant, soit X une variété projective sur K de dimension d ≥ 1. SoitLun diviseur sur X tel que q = h0 (X, L) ≥ 1. SoientD1,...,Dr des diviseurs effectifs
sur X qui se coupent proprement, tels que r i=1 Di soit non vide.
Introduisons quelques notations. Soit t ∈ . Pour x ∈ R+, ondéfinit l’idéal
J(t,x) de OX par
J(t,x) =
b∈N(t,x)
OX
−
r
biDi
i=1
et on pose F (t)x = Ɣ(X, J(t,x) ⊗ L). Observons que (F (t)x)x∈R+ est une filtration
de Ɣ(X, L). Pour tout s ∈ Ɣ(X, L) −{0}, on note µt(s) = sup{y ∈ R+ | s ∈ F (t)y}.
On pose finalement
F ( t ) = 1
+∞
h
q
0 X, J( t,x) ⊗ L dx.
Remarque 3.5
Soit B = (s1,...,sq) une base de Ɣ(X, L). On a alors l’inégalité
F ( t ) ≥ 1
q
+∞
0
0
# F ( t )x ∩ B dx = 1
q
,
q
µt(sk),
k=1

SUR LA NON-DENSITÉ DES POINTS ENTIERS 19
qui est même une égalité siB est adaptée à la filtration F ( t ).
Voici le résultat principal de cette section.
THÉORÈME 3.6
L’application F : → R+ ainsi définie est concave. On a en particulier F ( t ) ≥
mini α(L, Di) pour tout t ∈ .
Démonstration
Soient t ∈ , u ∈ et λ ∈ [0, 1]. Il s’agit de montrer que
F λt + (1 − λ)u ≥ λF ( t ) + (1 − λ)F ( u ).
Le lemme 3.1 construit une base B = (s1,...,sq) de W = Ɣ(X, L) adaptée aux
filtrations F ( t ) et F ( u ).
Soit (x,y) ∈ R2 + . Le lemme 3.3 (voir aussi remarque 3.4) fournit l’inclusion
d’idéaux J(t,x) ∩ J(u,y) ⊂ J(λt + (1 − λ)u,λx+ (1 − λ)y), puisque les diviseurs
D1,...,Dr se coupent proprement. On a en particulier l’inclusion d’espaces vectoriels
F ( t )x ∩ F ( u )y ⊂ F λt + (1 − λ) u
λx+(1−λ)y .
Soit s ∈ W −{0}. Ce qui précède implique que s appartient à F (λt + (1 −
λ)u)λx+(1−λ)y pour tout x

20 PASCAL AUTISSIER
Choisissons une section rationnelle s de L définie et non nulle en P . On pose
1
h ˆL(P ) =−
[K ′
ln s(P )v,
: Q]
où v parcourt l’ensemble des places de K ′ .Ceréel ne dépend pas du choix de s.
On va utiliser la version suivante du théorème du sous-espace.
PROPOSITION 4.2
On suppose L grand sur X. Notons q = h0 (X, L). Soient s1,...,sl des sections non
nulles engendrant Ɣ(X, L). Soit ε>0. Il existe alors un fermé Z = X tel que pour
toute extension finie K ′ de K et tout ensemble fini S de places de K ′ , l’ensemble des
points P ∈ (X − Z)(K ′ ) vérifiant
v∈S
max
J ∈L
j∈J
v
ln sj(P ) −1
v ≥ (q + qε)[K′ : Q]h ˆL(P ) (1)
est fini, où L désigne l’ensemble des parties J de {1,...,l} telles que (sj)j∈J soit une
base de Ɣ(X, L).
Démonstration
Posons P = P(Ɣ(X, L)).Désignons par π : X ′ → X l’éclatement de X relativement
à BL et par E = π −1 (BL) le diviseur exceptionnel. En notant L ′ = OX ′(−E) ⊗ π ∗L, on a un morphisme L ′ : X′ → P génériquement fini qui prolonge L. Il existe donc
un fermé Z1 = X ′ tel que L ′ |X ′ −Z1 soit à fibres finies.
On munit L0 = OP(1) d’une métrique adélique ( ′ v )v telle que la section globale
1E de ∗ L ′L∨0 ⊗ π ∗L vérifie
sup
X ′ 1E
v
′
v ≤ 1
pour toute place v. On applique alors la version de Vojta [14, théorème 0.3, reformulation
3.4] du théorème du sous-espace.
Il existe une réunion finie H de K-hyperplans de P P q−1
K telle que pour toute
extension finie K ′ de K et tout ensemble fini S de places de K ′ , l’ensemble des points
P ∈ (P − H )(K ′ ) vérifiant
ln sj(P ) ′−1
v
v∈S
max
J ∈L
j∈J
≥ (q + qε)[K′ : Q]hL0 ˆ (P )
est fini.
Prenons Z = BL ∪ π(Z1 ∪ −1
L ′ (H )). Alors pour toute extension finie K′ de K
et tout ensemble fini S de places de K ′ , l’ensemble des points P ∈ (X − Z)(K ′ )
vérifiant (1) est fini.

SUR LA NON-DENSITÉ DES POINTS ENTIERS 21
Montrons donc les critères de quasi-hyperbolicité delasection2.
Démonstration du théorème 2.11
On reprend en l’améliorant la démonstration du [1,théorème 3.3]. On procède en deux
étapes: dans la première, on construit un fermé Z = X candidat à contenir “presque
tous les points entiers”; dans la seconde, on prouve que Y est arithmétiquement quasihyperbolique.
Étape 1
Posons ε = (1/4m)(min α(L, Di) − m) et q = h
i 0 (X, L), et fixons un entier b ≥
(1/ε + 3)r. On choisit aussi une base B0 de Ɣ(X, L).
Désignons par P l’ensemble des parties I non vides de {1,...,r} telles que
i∈I Di soit non vide. Soit I ∈ P . On note I l’ensemble des a = (ai)i ∈ NI tels
que
i∈I ai = b. Soita∈I . Pour x ∈ R+, ondéfinit l’idéal J(x) de OX par
J(x) =
−
b
OX
où la somme porte sur les b ∈ NI tels que
aibi ≥ bx,
i∈I
i∈I
biDi
et on pose F (I,a)x = Ɣ(X, J(x) ⊗ L). Le lemme 3.1 fournit une base BI,a de
Ɣ(X, L) adaptée à la filtration F (I,a).
On munit chaque faisceau OX(Di) d’une métrique adélique. Appliquons le
théorème du sous-espace (proposition 4.2) avec {s1,...,sl} =B0 ∪
I,a BI,a (remarquons
que cette réunion est finie puisque P et les I le sont).
Il existe un fermé Z = X tel que pour toute extension finie K ′ de K et tout
ensemble fini S de places de K ′ , l’ensemble des points P ∈ (X − Z)(K ′ ) vérifiant
l’inégalité (1) est fini.
Étape 2
Soient K ′ une extension finie de K et S un ensemble fini de places de K ′ contenant
les places archimédiennes. Soit E ⊂ Y (K ′ ) un ensemble S-entier sur Y . Raisonnons
par l’absurde en supposant E − Z(K ′ ) infini. Choisissons une suite injective (Pn)n≥0
d’éléments de E − Z(K ′ ).
Quitte à extraire, on peut supposer (par compacité) que pour tout v ∈ S, la suite
(Pn)n≥0 converge dans X(K ′ v ) vers un yv ∈ X(K ′ v ).
Pour tout v ∈ S, on note Iv l’ensemble des i ∈{1,...,r} tels que yv ∈ Di.
Quitte à extraire de nouveau, on peut supposer que pour tout v ∈ S tel que Iv soit non

22 PASCAL AUTISSIER
vide et tout i ∈ Iv, la suite
ln 1Di(Pn)v
ln 1Dj (Pn)v
j∈Iv
converge vers un tvi ∈ [0, 1]. Observons que l’on a
i∈Iv tvi = 1.
Fait: Soitv∈S. Il existe une base (s1v,...,sqv) de Ɣ(X, L) contenue dans
{s1,...,sl} telle que l’on ait la minoration suivante pour tout n ≥ 0:
−
q
ln skv(Pn)v ≥−(q + 2qε)ln1L(Pn)v − O(1), (2)
k=1
où le O(1) est indépendant de n.
Prouvons ce fait. Si Iv est vide, on prend {s1v,...,sqv} =B0 et on obtient la
minoration (2) en remarquant que ln 1L(Pn)v = O(1) (puisque yv /∈ L).
On suppose maintenant Iv non vide. On a donc Iv ∈ P . Choisissons un a v =
(avi)i ∈ Iv tel que avi ≤ (b + r)tvi pour tout i ∈ Iv. On prend alors {s1v,...,sqv} =
BIv,a v .Vérifions que ce choix convient.
Soit s ∈ Ɣ(X, L) −{0}. Notons µ(s) le plus grand rationnel µ tel que s ∈
F (Iv,av )µ. Enécrivant localement s =
b fb
i∈Iv 1bi
Di au voisinage de yv, on
trouve
− ln s(Pn)v ≥−max
b
Par définition des tvi, ona
−bi ln 1Di(Pn)v ≥−
n≥0
bi ln 1Di(Pn)v − O(1).
i∈Iv
avibi
b + r
− mε
r
ln 1Dj (Pn)v
pour tout n assez grand et tout i ∈ Iv.
On en déduit l’inégalité (pour tout n ≥ 0)
µ(s)b
− ln s(Pn)v ≥− − mε ln 1Dj (Pn)v − O(1).
b + r
On écrit cette inégalité pour s = skv, puis on somme sur k. En observant que l’on a
q
µ(skv) =
k=1
j∈Iv
j∈Iv
+∞
dim F (Iv,av )x dx (remarque 3.5)
0
≥ min α(L, Di)q (théorème 3.6)
i
b + r
= (1 + 4ε)qm ≥ (q + 3qε)m,
b

SUR LA NON-DENSITÉ DES POINTS ENTIERS 23
on obtient alors
−
q
ln skv(Pn)v ≥−(q + 2qε)m
ln 1Dj (Pn)v − O(1).
k=1
Le fait énoncé (2) s’en déduit en remarquant que ln 1Dj (Pn)v = O(1) pour tout
j/∈ Iv.
Maintenant, l’ensemble E est S-entier sur Y , donc pour tout n ≥ 0, ona
[K ′ : Q]h ˆL(Pn) =−
ln 1L(Pn)v + O(1).
v∈S
j∈Iv
En utilisant la minoration (2), on trouve (pour tout n ≥ 0)
−
v∈S
q
ln skv(Pn)v ≥ (q + 2qε)[K ′ : Q]h ˆL(Pn) − O(1).
k=1
D’où une contradiction avec (1).
Démonstration du théorème 2.12
On reprend de la même manière la démonstration du [1, théorème 3.5], qui utilise
la version de Vojta [16, théorème 2] du théorème de Cartan au lieu du théorème du
sous-espace. Les détails sont laissés au lecteur.
5. Démonstration des théorèmes 1.3 et 1.4
Soient K un corps de nombres et X une variété projective sur K de dimension
d ≥ 2. Pour tous diviseurs L1,...,Ld sur X,ondésigne par
L1 ···Ld leur nombre
d’intersection.
Définition 5.1
Un diviseur L sur X est dit big lorsque lim inf
n→+∞ (1/nd )h 0 (X, nL) > 0.
Remarque 5.2
D’après le lemme de Kodaira [7, p. 141], le diviseur L est big si et seulement si nL
est grand pour tout entier n assez grand.
Définition 5.3
Soit L un diviseur big sur X. On dit que L est presque ample lorsqu’il existe un entier
n ≥ 1 tel que BnL soit vide.
On montre ici de légères extensions des théorèmes 1.3 et 1.4 de l’introduction.

24 PASCAL AUTISSIER
THÉORÈME 1.3BIS
Soient D1,...,D2d des diviseurs effectifs presque amples sur X qui se coupent
proprement. Alors X − D1 ∪···∪D2d est arithmétiquement quasi-hyperbolique.
Démonstration
D’après [1, théorème 4.4], il existe (m1,...,m2d) ∈ (N ∗ ) 2d tel qu’en posant L =
2d
i=1 miDi, onait
lim inf
n→+∞
1
n α(nL, miDi) > 1 pour tout i ∈{1,...,2d}.
On en déduit le résultat en appliquant le théorème 2.11 (les diviseurs
m1D1,...,m2dD2d se coupent encore proprement).
THÉORÈME 1.4BIS
Soient D1,...,Dr des diviseurs effectifs non nuls sur X qui se coupent proprement,
avec r ≥ d + 2. Posons L = r i=1 Di et Y = X − D1 ∪···∪Dr. On suppose
que L − (d + 1)Di est ample pour tout i ∈{1,...,r}.AlorsYest arithmétiquement
quasi-hyperbolique.
Démonstration
D’après le lemme 5.4 ci-dessous, on a lim inf
n→+∞ (1/n)α(nL, Di) > 1 pour tout i ∈
{1,...,r}. On obtient le résultat en appliquant le théorème 2.11.
LEMME 5.4
Soient L un diviseur ample sur X et D un diviseur effectif non nul sur X. Soit t ≥ 1
un entier tel que L − tD soit ample. On a alors les minorations
lim inf
n→+∞
1
α(nL, D) ≥
n
t
(d + 1) L d
d
j=0
L d−j (L − tD) j > t
d + 1 .
Démonstration
La formule de Hirzebruch-Riemann-Roch donne que χ(X, nL−kD) est une fonction
polynomiale en (n, k) dont on peut expliciter la composante homogène dominante.
Pour tout (n, k) tel que 0 ≤ k ≤ tn,onaχ(X, nL − kD) = (1/d!)〈(nL −
kD) d 〉+O(n d−1 ).
Par ailleurs, pour tout n assez grand et tout k ∈{0,...,tn}, onah i (X, nL −
kD) = 0 pour tout i ≥ 1, puisque L et L − tD sont amples. On a en particulier
h 0 (X, nL − kD) = (1/d!)〈(nL − kD) d 〉+O(n d−1 ).

SUR LA NON-DENSITÉ DES POINTS ENTIERS 25
On trouve ainsi les estimations suivantes:
tn
k=1
h 0 (X, nL − kD) = 1
d!
= 1
d!
= 1
d!
tn
k=1
d
j=0 k=1
d
j=0
〈(nL − kD) d 〉+O(n d−1 )
tn
Or un calcul montre (par multilinéarité) la formule
d
j=0
C j (−1) j
d−j j
d L D
j + 1 t j+1 = t
d + 1
C j
d 〈Ld−j D j 〉n d−j (−k) j + O(n d )
C j
d 〈Ld−j D j 〉 (−1)j
j + 1 t j+1 n d+1 + O(n d ).
d
〈L d−j (L − tD) j 〉.
D’où la première inégalité del’énoncé.
La deuxième inégalité s’en déduit facilement: les diviseurs L et L − tD sont
amples, donc on a 〈L d−j (L − tD) j 〉 > 0 pour tout j ∈{1,...,d}.
6. Démonstration du lemme 3.3
Soit A un anneau local noethérien. On aura besoin d’un résultat classique dû à Krull.
LEMME 6.1
Soient I et J deux idéaux de A avec J = A. On a alors l’égalité
(I + J k ) = I.
k≥1
Démonstration
Il suffit d’appliquer [9, corollaire 11.D.2, p. 69], dans l’anneau A/I.
Soit (ϕ1,...,ϕr) une suite régulière de A. Rappelons que pour toute partie saturée N
de Nr ,ondésigne par I(N) l’idéal 〈ϕ b1
1 ···ϕbr r ,b∈ N〉 de A. Le lemme 3.3 est une
conséquence directe de l’énoncé suivant.
LEMME 6.2
Soit N une partie saturée de N r . On a alors l’égalité d’idéaux
Démonstration
Notons ici I1(N) =
I(N) =
c∈N r −N
〈ϕ c1+1
1
c∈Nr −N 〈ϕc1+1 1 ,...,ϕcr +1
r
j=0
,...,ϕ cr +1
r
〉.
〉. Vérifions d’abord l’inclusion

26 PASCAL AUTISSIER
I(N) ⊂ I1(N):sib ∈ N et c ∈ Nr − N, alors il existe un indice i tel que bi ≥ ci + 1,
donc ϕ b1
1 ,...,ϕbr r ∈ ϕci+1 i A ⊂〈ϕc1+1 1 ,...,ϕcr +1
r 〉.
On prouve l’inclusion I1(N) ⊂ I(N) par récurrence sur r. Le cas r = 1 est
facile (avec le lemme 6.1 si N est vide). Supposons r ≥ 2 et le résultat au cran r − 1.
Montrons par une (deuxième) récurrence sur k que I1(N) ⊂ I(N) + ϕk 1A pour tout
k ∈ N. Sik = 0, cette inclusion est évidente. Supposons k ≥ 1 et le résultat au cran
k − 1. Soitsun élément de I1(N). Ils’écrit s = x + ϕ k−1
1 a avec un x ∈ I(N) et un
a ∈ A.
Posons Nk = {b = (b2,...,br) ∈ Nr−1 | (k − 1,b2,...,br) ∈ N}. Soit
b ∈ N r−1 − Nk. Alorsϕ k−1
1 a = s − x appartient à I1(N) donc en particulier à
〈ϕk 1 ,ϕb2+1 2 ,...,ϕbr +1
r 〉. Autrement dit, il existe a ′ ∈ A tel que ϕ k−1
1 a − ϕk 1a′ ∈
〈ϕ b2+1
2 ,...,ϕbr +1
r 〉. Puisque (ϕ b2+1
2 ,...,ϕbr +1
r ,ϕ1) est une suite régulière de A, on
obtient a − ϕ1a ′ ∈〈ϕ b2+1
2 ,...,ϕbr +1
r 〉 en simplifiant par ϕ k−1
1 .Réduisons modulo ϕ1
on trouve dans A ′ = A/ϕ1A que ā ∈〈¯ ϕ2 b2+1 ,..., ϕr ¯ br +1 〉.
On a ainsi que ā ∈
b∈Nr−1 〈 ¯ −Nk ϕ2 b2+1 ,..., ϕr ¯ br +1 〉.Or(¯ ϕ2,..., ϕr) ¯ est une suite
régulière de A ′ , donc ā ∈〈¯ ϕ2 b2 ··· ϕr ¯ br , b ∈ Nk〉 par la première hypothèse de
récurrence. On écrit a = y + ϕ1a1 avec y ∈〈ϕ b2
2 ···ϕbr r ,b∈ Nk〉 et a1 ∈ A. Ilen
découle l’égalité s = x + ϕ k−1
1 y + ϕk 1a1 avec ϕ k−1
1 y ∈〈ϕk−1 1 ϕb2 2 ···ϕbr r ,b∈ Nk〉 ⊂
I(N), qui implique bien que s est dans I(N) + ϕk 1A. D’où l’inclusion I1(N) ⊂
k∈N (I(N) + ϕk 1A). On conclut par le lemme 6.1.
Remerciements. Je remercie Yuri Bilu pour m’avoir fourni la référence [2]. Je remercie
également les rapporteurs pour leurs suggestions pertinentes.
References
[1] P. AUTISSIER, Géométrie, points entiers et courbes entières, Ann. Sci. Éc. Norm.
Supér. (4) 42 (2009), 221 – 239. MR 2518077 14, 17, 21, 23, 24
[2] P. CORVAJA, A. LEVIN et U. ZANNIER, Integral points on threefolds and other
varieties, Tohoku Math. J. 61 (2009), 589 – 601. MR 2598251 14, 26
[3] P. CORVAJA et U. ZANNIER, A subspace theorem approach to integral points on curves,
C. R. Math. Acad. Sci. Paris 334 (2002), 267 – 271. MR 1891001 14
[4] ———, On integral points on surfaces, Ann. of. Math. (2) 160 (2004), 705 – 726.
MR 2123936 14, 17
[5] W. FULTON, Intersection Theory, 2nd ed., Ergeb. Math. Grenzgeb. 3, Springer, Berlin,
1998. MR 1644323 16
[6] S. LANG, Number Theory, III: Diophantine Geometry, Encyclopaedia Math. Sci. 60,
Springer, Berlin, 1991. MR 1112552 13

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[7] R. LAZARSFELD, Positivity in Algebraic Geometry, I: Classical Setting—Line Bundles
and Linear Series, Ergeb. Math. Grenzgeb. 48, Springer, Berlin, 2004.
MR 2095471 23
[8] A. LEVIN, Generalizations of Siegel’s and Picard’s theorems, Ann. of Math. (2) 170
(2009), 609 – 655. MR 2552103 14, 17
[9] H. MATSUMURA, Commutative Algebra, 2nd ed., Math. Lecture Note Ser. 56,
Benjamin/Cummings, Reading, Mass., 1980. MR 0575344 25
[10] H. P. SCHLICKEWEI, The p-adic Thue-Siegel-Roth-Schmidt theorem, Arch. Math.
(Basel) 29 (1977), 267 – 270. MR 0491529 14
[11] W. M. SCHMIDT, Diophantine approximation, Lecture Notes in Math. 785, Springer
Berlin, 1980. MR 0568710 14
[12] J.-P. SERRE, Lectures on the Mordell-Weil Theorem, 3rd ed., Aspects Math., Friedr.
Vieweg and Sons, Braunschweig, 1997. MR 1757192
[13] P. VOJTA, Diophantine Approximations and Value Distribution Theory, Lecture Notes
in Math. 1239, Springer, Berlin, 1987. MR 0883451 13, 15
[14] ———, A refinement of Schmidt’s subspace theorem, Amer. J. Math. 111 (1989),
489 – 518. MR 1002010 20
[15] ———, Integral points on subvarieties of semiabelian varieties, I, Invent. Math. 126
(1996), 133 – 181. MR 1408559 14
[16] ———, On Cartan’s theorem and Cartan’s conjecture, Amer. J. Math. 119 (1997),
1 – 17. MR 1428056 23
[17] S. ZHANG, Small points and adelic metrics,J.AlgebraicGeom.4 (1995), 281 – 300.
MR 1311351 19
Institut de Mathématiques de Bordeaux, Université Bordeaux I, 33405 Talence CEDEX,
France; pascal.autissier@math.u-bordeaux1.fr

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS
CASIM ABBAS
Abstract
Giroux showed that every contact structure on a closed 3-dimensional manifold is
supported by an open book decomposition. We extend this result by showing that the
open book decomposition can be chosen in such a way that the pages are solutions to
a homological perturbed holomorphic curve equation.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2. Existence and local foliations . . . . . . . . . . . . . . . . . . . . . . . 36
3. From local foliations to global ones . . . . . . . . . . . . . . . . . . . 52
4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Appendix. Some local computations near the punctures . . . . . . . . . . . 78
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
1. Introduction
This paper is the starting point of a larger program by the author, Hofer, and Lisi
investigating a perturbed holomorphic curve equation in the symplectization of a 3dimensional
contact manifold (see [6], [7]). One aim of this program is to provide
an alternative proof of the Weinstein conjecture in dimension 3 as outlined in [4]
complementing Taubes’s gauge theoretical proof (see [33], [34]). A special case of
this paper’s main result has been used in the proof of the Weinstein conjecture for
planar contact structures in [4]. Another reason for studying this equation is to construct
foliations by surfaces of section with nontrivial genus. This is usually impossible to
do with the unperturbed holomorphic curve equation since solutions generically do
not exist.
Consider a closed 3-dimensional manifold M equipped with a contact form λ.
This is a 1-form which satisfies λ ∧ dλ = 0 at every point of M. We denote the
associated contact structure by ξ = ker λ, and we denote the Reeb vector field by Xλ.
DUKE MATHEMATICAL JOURNAL
Vol. 158, No. 1, c○ 2011 DOI 10.1215/00127094-1276301
Received 20 July 2009. Revision received 8 July 2010.
2010 Mathematics Subject Classification. Primary 53D35; Secondary 35J62.
29

30 CASIM ABBAS
Recall that the Reeb vector field is defined by the two equations
iXλdλ = 0 and iXλλ = 1.
Definition 1.1 (Open book decomposition)
Assume that K ⊂ M is a link in M and that τ : M\K → S 1 is a fibration so
that the fibers Fϑ = τ −1 (ϑ) are interiors of compact embedded surfaces ¯Fϑ with
∂ ¯Fϑ = K, where ϑ is the coordinate along K. We also assume that K has a tubular
neighborhood K × D, D ⊂ R 2 being the open unit disk, such that τ restricted to
K × (D\{0}) is given by τ(ϑ, r, φ) = φ, where (r, φ) are polar coordinates on D.
Then we call τ an open book decomposition of M, the link K is called the binding of
the open book decomposition, and the surfaces Fϑ are called the pages of the open
book decomposition.
It is a well-known result in 3-dimensional topology that every closed 3-dimensional
orientable manifold admits an open book decomposition. Indeed, Alexander proved
the following theorem in 1923 (see [10], [30]):
THEOREM 1.2
Every closed, orientable manifold M of dimension 3 is diffeomorphic to
W (h) ∪Id (∂W × D 2 ),
where D 2 is the closed unit disk in R 2 ,whereW is an orientable surface with boundary,
and where h : W → W is an orientation-preserving diffeomorphism which restricts
to the identity near ∂W.HereW (h) denotes the manifold obtained from W × [0, 2π]
by identifying (x,0) with (h(x), 2π).
The above decomposition is an open book decomposition, and the pages are given by
Fϑ := (W ×{ϑ}) ∪Id (∂W × Iϑ), 0 ≤ ϑ

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 31
dλ ′ induces an area form on each fiber Fϑ with K consisting of closed orbits of the
Reeb vector field Xλ ′,andλ′ orients K as the boundary of (Fϑ,dλ ′ ).
We refer to a contact form λ ′ above as a Giroux contact form. Note that λ ′ is not unique
and that it is in general different from the original contact form λ. The following
theorem by Giroux guarantees existence of such open book decompositions, and it
contains a uniqueness statement as well (see also [16, Proposition 2]).
THEOREM 1.4 ([16, Theorem 3])
Every co-oriented contact structure ξ = ker λ on a closed 3-dimensional manifold is
supported by some open book. Conversely, if two contact structures are supported by
the same open book, then they are diffeomorphic.
In the topological category, it is possible to modify an open book decomposition
such that the pages of the new decomposition have lower genus at the expense of
increasing the number of connected components of K. It was not known for some
time whether a similar statement could also be made in the context of supporting open
book decompositions. In particular, it was unclear whether every contact structure
was supported by an open book decomposition whose pages were punctured spheres
(planar pages). The author and his collaborators could resolve the Weinstein conjecture
for contact forms inducing a planar contact structure in 2005 (see [4]). So the question
of whether all contact structures are planar became a priority, which prompted Etnyre
to address it in [14]. He showed that overtwisted contact structures always admit
supporting open book decompositions with planar pages, but many contact structures
do not. Since then, planar open book decompositions have become an important tool
in contact geometry.
In this paper, we will prove that every contact structure has a supporting open book
decomposition such that the pages solve a homological perturbed Cauchy-Riemann
type equation which we now describe after introducing some notation. We write
πλ = π : TM → ξ for the projection along the Reeb vector field Xλ. Fix a complex
multiplication J : ξ → ξ so that the map ξ ⊕ ξ → R, definedby
(h, k) → dλ(h, J k),
defines a positive definite metric on the fibers. We call such complex multiplications
compatible (with dλ). The equation of interest here is the following nonlinear firstorder
elliptic system. The solutions consist of 5-tuplets (S,j,Ɣ,ũ, γ ) where (S,j)
is a closed Riemann surface with complex structure j, Ɣ ⊂ S is a finite subset,
ũ = (a,u) : ˙S → R × M is a proper map with ˙S = S \ Ɣ,andγ is a 1-form on S so

32 CASIM ABBAS
that
Here the energy E(ũ) is defined by
⎧
⎪⎨
π ◦ Tu◦ j = J ◦ π ◦ Tu on ˙S
(u
⎪⎩
∗λ) ◦ j = da + γ on ˙S
dγ = d(γ ◦ j) = 0 on S
E(ũ) < ∞.
E(ũ) = sup
ϕ∈
˙S
ũ ∗ d(ϕλ),
(1.1)
where consists of all smooth maps ϕ : R → [0, 1] with ϕ ′ (s) ≥ 0 for all s ∈ R.
Note that equation (1.1) reduces to the usual pseudoholomorphic curve equation
in the symplectization R × M if we set γ = 0. The following proposition, which is a
modification of a result by Hofer [17], shows that solutions to problem (1.1) approach
cylinders over periodic orbits of the Reeb vector field.
PROPOSITION 1.5
Let (M,λ) be a closed 3-dimensional manifold equipped with a contact form λ.Then
the associated Reeb vector field has periodic orbits if and only if the associated PDE
problem (1.1) has a nonconstant solution.
Proof
Let (S,j,Ɣ,ũ, γ ) be a nonconstant solution of (1.1). If Ɣ = ∅, then the results in [17]
imply that, near a puncture, the solution is asymptotic to a periodic orbit (see also [3]
for a complete proof). Here we use the fact that γ is exact near the punctures. The
aim now is to show that, in the absence of punctures, the map a is constant while the
image of u lies on a periodic Reeb orbit. Assume that Ɣ =∅.Since
we find, after applying d, that
u ∗ λ =−da ◦ j − γ ◦ j,
ja =−d(da ◦ j) = u ∗ dλ.
In view of the equation π ◦ Tu◦ j = J ◦ π ◦ Tu, we see that u ∗ dλ is a nonnegative
integrand. Applying Stokes’s theorem, we obtain
S u∗ dλ = 0, implying that
π ◦ Tu≡ 0.

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 33
Hence a is a harmonic function on S and therefore constant. So far, we also know that
the image of u lies on a Reeb trajectory, and it remains to show that this trajectory is
actually periodic.
Let τ : ˜S → S be the universal covering map. The complex structure j lifts to a
complex structure ˜j on ˜S. Now pick smooth functions f, g on ˜S such that
dg = τ ∗ γ =: ˜γ, −df = τ ∗ (γ ◦ j) = ˜γ ◦ ˜j.
Then the map u ◦ τ : ˜S → M satisfies
(u ◦ τ) ∗ λ = df.
Theimageofu◦ τ lies on a trajectory x of the Reeb vector field in view of
D(u ◦ τ)(z)ζ = Df (z)ζ · Xλ (u ◦ τ)(z) ,
hence (u ◦ τ)(z) = x(h(z)) for some smooth function h on ˜S, and it follows that, after
maybe adding a constant to f ,wehave
(u ◦ τ)(z) = x f (z) .
The function f does not descend to S. If it did, it would have to be constant since it
is harmonic. On the other hand, this would imply that u is constant in contradiction to
our assumption that it is not. Therefore, there is a point q ∈ S and two lifts z0,z1 ∈ ˜S
such that f (z0) >f(z1).Letℓ : S 1 → S be a loop which lifts to a path α :[0, 1] → ˜S
with α(0) = z0 and α(1) = z1. Considering the map
v := u ◦ ℓ : S 1 −→ M,
we see that v(t) = (u ◦ τ ◦ α)(t) = x f (α(t)) and x(f (z0)) = x(f (z1)), that is, the
trajectory x is a periodic orbit. Hence the image of u is a periodic orbit for the Reeb
vector field.
The following is the main result of this paper.
THEOREM 1.6
Let M be a closed 3-dimensional manifold, and let λ ′ be a contact form on M. Then
the following holds for a suitable contact form λ = fλ ′ ,wheref is a positive function
on M. There exists a smooth family (S,jτ ,Ɣτ , ũτ = (aτ ,uτ ),γτ )τ∈S 1 of solutions to
(1.1) for a suitable compatible complex structure J :kerλ → ker λ such that
• all maps uτ have the same asymptotic limit K at the punctures, where K is a
finite union of periodic trajectories of the Reeb vector field Xλ;

34 CASIM ABBAS
• uτ ( ˙S) ∩ uτ ′( ˙S) =∅ if τ = τ ′ ;
• M\K =
τ∈S 1 uτ ( ˙S);
• the projection P onto S 1 defined by p ∈ uτ ( ˙S) ↦→ τ is a fibration;
• the open book decomposition given by (P,K) supports the contact structure
ker λ, and λ is a Giroux form.
Here is a very brief outline of the argument. The reader is invited to skip forward to
Section 4 to see in more detail how all the partial results of this paper are tied together
to prove the main result. In Section 2, we find a Giroux contact form which has a
certain normal form near the binding. Following an argument by Wendl ([39], [38]),
we will then almost be able to turn the Giroux leaves into solutions of (1.1) without
harmonic form except for the fact that we have to accept a confoliation form instead
of a contact form. Pick one of these Giroux leaves as a starting point. The next step is
to prove a result which permits us to perturb the Giroux leaf into a genuine solution
of (1.1) while simultaneously perturbing the confoliation form slightly into a contact
form. This is where the harmonic form in (1.1) is required. We actually obtain a local
family of nearby solutions, not just one. In Section 3, we prove a compactness result
which extends the local family of solutions into a global one. The remarkable fact is
that there is a compactness result in the context of this paper, although there is none
in general for the perturbed holomorphic curve equation. The special circumstances
in this paper imply a crucial a priori bound which implies that a sequence of solutions
has a pointwise convergent subsequence with a measurable limit. The objective is then
to show that the regularity of this limit is much better, that it is actually smooth.
We consider two solutions (S,j,Ɣ,ũ, γ ) and (S ′ ,j ′ ,Ɣ ′ , ũ ′ ,γ ′ ) equivalent if there
exists a biholomorphic map φ :(S,j) → (S ′ ,j ′ ) mapping Ɣ to Ɣ ′ (preserving the
enumeration) so that ũ ′ ◦ φ = ũ. We will often identify a solution (S,j,Ɣ,ũ, γ ) of
(1.1) with its equivalence class [S,j,Ɣ,ũ, γ ].WenotethatwehaveanaturalR-action
on the solution set by associating to c ∈ R and [S,j,Ɣ,ũ, γ ] the new solution
c + [S,j,Ɣ,ũ, γ ] = [S,j,Ɣ,(a + c, u),γ], ũ = (a,u).
A crucial concept for our discussion is the notion of a finite energy foliation F .
Definition 1.7 (Finite energy foliation)
A foliation F of R × M is called a finite energy foliation if every leaf F is the image
of an embedded solution [S,j,Ɣ,ũ, γ ] of the equations (1.1), that is,
F = ũ( ˙S),

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 35
so that u( ˙S) ⊂ M is transverse to the Reeb vector field, and for every leaf F ∈ F we
also have c + F ∈ F for every c ∈ R; that is, the foliation is R-invariant.
We recall the concept of a global surface of section. Let M be a closed 3-manifold,
and let X be a nowhere-vanishing smooth vector field.
Definition 1.8 (Surface of section)
(a) A local surface of section for (M,X) consists of an embedded compact surface
⊂ M with boundary, so that ∂ consists of a finite union of periodic orbits
(called the binding orbits). In addition, the interior ˙ = \ ∂ is transverse
to the flow.
(b) A local surface of section is called a global surface of section if, in addition,
every orbit other than a binding orbit hits ˙ in forward and backward time.
Furthermore, the globally defined return map : ˙ → ˙ has a bounded
return time; that is, there exists a constant c>0 so that every x ∈ ˙ hits ˙
again in forward time not exceeding c.
Using Proposition 2.5 below, the existence part of Giroux’s theorem can be rephrased
as follows.
THEOREM 1.9
Let M be a closed orientable 3-manifold, and let ˜λ be a contact form on M. Then
there exists a smooth function f : M → (0, ∞) so that the contact form λ = f ˜λ has
a Reeb vector field admitting a global surface of section.
Existence results for finite energy foliations with a given contact form λ are hard to
come by since they usually have striking consequences. In [20] for example, Hofer,
Wysocki, and Zehnder show that every compact strictly convex energy hypersurface
S in R 4 carries either two or infinitely many closed characteristics. The proof relies on
constructing a special finite energy foliation. In special cases they were established by
Hofer, Wysocki, and Zehnder [22] and by Wendl ([38], [40]). Proofs usually require
a starting point, that is, a finite energy foliation for a slightly different situation as the
given one. Then some kind of continuation argument is employed where all kinds of
things can and do happen to the original foliation. In [22], the authors start with an
explicit finite energy foliation for the round 3-dimensional sphere S 3 ⊂ R 4 which is
then deformed. Wendl’s papers also use a rather special manifold as a starting point.
The main result of this paper, Theorem 1.6, provides a starting finite energy foliation
for any closed 3-dimensional contact manifold (M,ker λ) since it is obtained from
deforming the leaves of Giroux’s open book decomposition. The pages are usually
not punctured spheres, and generically there are no pseudoholomorphic curves on

36 CASIM ABBAS
punctured surfaces with genus which are transverse to the Reeb vector field. This
makes the introduction of the harmonic form in (1.1) a necessity. The price to be paid
is that compactness issues are more complicated.
Wendl [39] published a proof of Theorem 1.6 for the special case where ker λ is
a planar contact structure, that is, where the surfaces ˙S are punctured spheres. This
result was outlined in [4]. Regardless of whether the contact structure is planar or not,
there are two main steps in the proof: existence of a solution and compactness of a
family of solutions. While the author established the compactness part for Theorem
1.6 long before [4] appeared, we will use the same argument described by Wendl in
[39] for the existence part since it simplifies the proof considerably.
The main theorem of this article was the first step in the proof of the Weinstein
conjecture for the planar case in [4]. Recall that the Weinstein conjecture [38, p. 358]
states the following: Every Reeb vector field X on a closed contact manifold M admits
a periodic orbit.
In fact, Weinstein added the additional hypothesis that the first cohomology group
H 1 (M,R) with real coefficients vanishes, but there seems to be no indication that this
additional hypothesis is needed.
Moreover, Theorem 1.6 is also the starting point for the construction of global
surfaces of section in the forthcoming paper [7]. Another application will be an
alternative proof of the Weinstein conjecture in dimension 3 (see [7]) as outlined in
[4]. This complements Taubes’s recent proof of the Weinstein conjecture in dimension
3 using a perturbed version of the Seiberg-Witten equations (see [33], [34]). The
main issue with the homological perturbed holomorphic curve equation (1.1) isthat
there is no natural compactification of the space of solutions unless the harmonic
forms are uniformly bounded. In the forthcoming papers [6] and[7] the lack of compactness
is investigated, and bounds for the harmonic forms are derived in particular
cases.
2. Existence and local foliations
2.1. Local model near the binding orbits
We use the same approach as in [39] and[38] to prove existence of a solution to
(1.1). Given a closed contact 3-manifold (M,ξ), Giroux’s theorem implies that there
is an open book decomposition as in Theorem 1.2 supporting ξ. On the other hand,
any other contact structure ξ ′ supported by the same open book is diffeomorphic to
ξ. Starting with an open book decomposition for M, we construct a contact structure
supported by Giroux contact form λ which has a certain normal form near the
binding.

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 37
Definition 2.1
Let θ ∈ S 1 = R/2πZ denote polar coordinates on the unit disk D ⊂ R 2 by (r, φ),
and let γ1,γ2 :[0, +∞) → R be smooth functions. A 1-form
λ = γ1(r) dθ + γ2(r) dφ
is called a local model near the binding if the following conditions are satisfied:
(1) the functions γ1,γ2,andγ2(r)/r 2 are smooth if considered as functions on the
disk D (in particular, γ ′ 1 (0) = γ ′ 2 (0) = γ2(0) = 0);
(2) µ(r) := γ1(r)γ ′ 2 (r) − γ ′ 1 (r)γ2(r) > 0 if r>0;
(3) γ1(0) > 0 and γ ′ 1 (r) < 0 if r>0;
(4) limr→0(µ(r)/r) = γ1(0)γ ′′
2 (0) > 0;
(5) κ := (γ ′′ ′′
1 (0)/γ 2 (0)) /∈ Z and κ ≤−1/2;
(6) A(r) = (1/µ 2 (r))(γ ′′
2 (r)γ ′ ′′
1 (r) − γ 1 (r)γ ′ 2
We explain some of the conditions above. First, since
(r)) is of order r for small r>0.
λ ∧ dλ = µ(r)dθ ∧ dr ∧ dφ = µ(r)
dθ ∧ dx ∧ dy,
r
the form λ is a contact form on S 1 × D. The Reeb vector field is given by
X(θ,r,φ) = γ ′ 2 (r)
µ(r)
∂
∂θ − γ ′ 1 (r)
µ(r)
∂
∂φ
The trajectories of X all lie on tori Tr = S 1 × ∂Dr:
We compute
and
=: α(r) ∂
∂θ
+ β(r) ∂
∂φ .
θ(t) = θ0 + α(r) t, φ(t) = φ0 + β(r) t. (2.1)
γ
lim α(r) = lim
r→0 r→0
′′
2 (r)
µ ′ (r)
γ
lim β(r) =−lim
r→0 r→0
′′
1 (r)
µ ′ (r)
= γ ′′
2 (0)
Recalling that ∂ ∂ ∂
= x − y , we obtain for r = 0
∂φ ∂y ∂x
X = 1 ∂
γ1(0) ∂θ ,
γ1(0)γ ′′
1
=
(0) γ1(0)
2
=− γ ′′
1 (0)
γ1(0)γ ′′
2 (0).

38 CASIM ABBAS
that is, the central orbit has minimal period 2πγ1(0). If the ratio α(r)/β(r) is irrational
then the torus Tr carries no periodic trajectories. Otherwise, Tr is foliated with periodic
trajectories of minimal period
τ = 2πm
α
= 2πn
β ,
where α/β = m/n or β/α = n/m for suitable integers m, n (choose whatever makes
sense if either α or β is zero). We calculate
dα
lim
r→0 dr
γ
= lim
r→0
′′
2 (r)µ(r) − γ ′ 2 (r)µ′ (r)
µ 2 (r)
γ
= lim
r→0
′′′
2 (r)
2µ ′ − lim
(r) r→0
= γ ′′′
= 0
since µ ′′ (0) = γ1(0)γ ′′′
coordinates on the disk, we get
2 (0)
2µ ′ (0)
µ ′′ (r)
2µ ′ γ
(r)
′ 2 (r)
r
′′′ ′′
γ1(0)γ 2 (0)γ 2
− (0)
2(µ ′ (0)) 2
2 (0) and µ′ (0) = γ1(0)γ ′′
2
X(θ,x,y) = α(x,y) ∂
∂θ
− β(x,y) y ∂
∂x
r
µ(r)
(0) > 0. Converting to Cartesian
+ β(x,y) x ∂
∂y ,
and linearizing the Reeb vector field along the center orbit yields
⎛
0 0 0
⎞
DX(θ,0, 0) = ⎝ 0 0 −β(0) ⎠ .
0 β(0) 0
The linearization of the Reeb flow is given by
⎛
1 0 0
⎞
Dφt(θ,0, 0) = ⎝ 0 cos β(0)t − sin β(0)t ⎠ (2.2)
0 sin β(0)t cos β(0)t
with
The spectrum of (t) is given by
(t) = e β(0)tJ , J =
σ (t) ={e ±iβ(0)t }.
0 −1
.
1 0

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 39
The binding orbit has period 2πγ1(0), since
and it is nondegenerate and elliptic.
′′ γ 1
γ1(0)β(0) =− (0)
γ ′′
2
(0) /∈ Z,
Example 2.2
For the contact form Tdθ+(1/k)(xdy−ydx) = Tdθ+(r 2 /k)dφ the central orbit
S 1 ×{0} is degenerate, but
is a local model near the binding if
In this case,
and we note that
If
λ = (1 − r 2 )(T dθ+ r2
k dφ)
k, T > 0, kT /∈ Z, and kT ≥ 1
2 .
µ(r) = 2rT
k (1 − r2 ) 2 > 0 and
A(r) = 1
µ 2 (r)
γ ′′
2
(r)γ ′
1
α(r)
β(r) =−γ ′ 2 (r)
γ ′ 1
′′ ′
(r) − γ 1 (r)γ 2 (r) =
1 − 2r2
=
(r) kT
γ ′′
1 (0)
γ ′′
2
m
=
n
(0) =−kT,
4kr
T (1 − r2 .
) 4
for integers n, m, then the invariant torus Tr is foliated with periodic orbits. The
case m = 0 is only possible if r = 1/ √ 2.Ifr is sufficiently small, then |m| ≥2.
Indeed, we would otherwise be able to find sequences rl ↘ 0 and {nl} ⊂Z such that
kT/(1 − 2r 2 l ) = nl, which is impossible. The binding orbit has period 2πT while
the periodic orbits close to the binding orbit have much larger periods equal to
τ = 2πTm (1 − r2 ) 2
.
1 − 2r2 Example 2.3
Consider the contact form λ = T (1 − r 2 )dθ + (r 2 /k)dφ on S 1 × D. Itisalsoa
local model near the binding if k, T > 0, kT ≥ 1/2, andkT is not an integer. We

40 CASIM ABBAS
even have A(r) ≡ 0. In contrast to Example 2.2, if kT /∈ Q, then the invariant tori Tr
carry no periodic orbits. If kT = n/m ∈ Q but not in Z, all invariant tori are foliated
with periodic orbits of period 2πmT with |m| ≥2 while the binding orbit has period
2πT. The function A(r) is identically zero. This is the contact form on the irrational
ellipsoid in R 4 .
The following proposition is essentially due to Wendl. The construction in the proof
was used by Thurston and Winkelnkemper [35] to show existence of contact forms on
closed 3-manifolds.
PROPOSITION 2.4 ([39, Proposition 1])
Let M be a 3-dimensional manifold given by an open book decomposition
M = W (h) ∪Id (∂W × D 2 )
as described in Theorem 1.2. We denote the pages by
Fα := (W ×{α}) ∪Id (∂W × Iα), 0 ≤ α

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 41
([0,ε] × ( ˙
nS1 ), {0} ×( ˙
nS1 )), where we take an n-fold disjoint union of circles
S1 ≈ R/2πZ according to the number n of components of ∂W.
We
claim that there is an area form on W that satisfies
• = 2πn,
W
• |C = dt ∧ dθ.
Indeed, start with any area form ′ so that
W ′ = 2πn.Thenwehave ′ |C =
f ′ (t,θ)dt∧dθ with a positive smooth function f ′ (after switching signs if necessary).
Now pick a new smooth positive function f which is equal to some constant c if
t ≤ (1/3)ε and which agrees with f ′ if t ≥ (2/3)ε so that the resulting area form
still satisfies
= 2πn. Do one component of ∂W at a time. Rescaling the
W
t-coordinate, we may assume that c = 1.
Let α1 be any 1-form on W which equals (1 + t) dθ near ∂W. Then by Stokes’s
theorem we obtain
( − dα1) = 2πn− α1 = 2πn+ dθ = 0.
W
∂W
∂W
The 2-form −dα1 on W is closed and vanishes near ∂W. Then there exists a 1-form
β on W with
dβ = − dα1
and β ≡ 0 near ∂W.Nowdefineα2 := α1 + β.Thenα2 satisfies the following:
• dα2 is an area form on W inducing the same orientation as , (2.3)
• α2 = (1 + t) dθ near ∂W. (2.4)
The set of 1-forms on W satisfying (2.3) and (2.4) is therefore nonempty and also
convex. We define the following 1-form on W × [0, 2π], where α is any 1-form on
W satisfying (2.3) and (2.4):
˜α(x,τ):= τα(x) + (2π − τ)(h ∗ α)(x).
This 1-form descends to the quotient W (h), and the restriction to each fiber of the
fiber bundle W (h) π → S 1 satisfies condition (2.3). Moreover, since h ≡ Id near ∂W,
we have ˜α(x,τ) = 2π(1 + t) dθ for all (x,τ) = ((t,θ),τ) near ∂W(h) = ∂W × S 1 .
Let dτ be a volume form on S 1 . We claim that
λ1 := −δ ˜α + π ∗ dτ

42 CASIM ABBAS
are contact forms on W (h) whenever δ>0 is sufficiently small. Pick (x,τ) ∈ W(h),
and let {u, v, w} be a basis of T(x,τ)W (h) with π∗u = π∗v = 0. Then
(λ1 ∧ dλ1)(x,τ)(u, v, w)
= δ 2 (˜α ∧ d ˜α)(x,τ)(u, v, w) − δ [dτ(π∗w) d ˜α(x,τ)(u, v)]
= 0
for sufficiently small δ>0,anddλ1 is a volume form on W . Now we have to continue
the contact forms λ1 beyond ∂W(h) ≈ ∂W × S1 onto ∂W × D2 . At this point it is
convenient to change coordinates. We identify C × S1 with ∂W × (D2 1+ε \D2 1 ), where
D2 ρ is the 2-disk of radius ρ. Using polar coordinates (r, φ) on D2 1+ε with 0 ≤ φ ≤ 2π
and 0 0 and γ ′
1 (r) < 0 if r>0,
hence the curves γ = γδ have to turn counterclockwise in the first quadrant starting
at the point (γ1(0), 0) and later connecting with (−δ(1 − ε0), 1). In the case where
δ>0, the Reeb vector fields are given by
Xδ(θ,r,φ) = γ ′ 2 (r)
µ(r)
∂
∂θ − γ ′ 1 (r)
µ(r)
∂
∂φ ,

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 43
and, in particular,
Xδ(θ,r,φ) = ∂
∂φ for r ≥ 1 − ε0, (2.5)
which implies that the Reeb vector fields Xδ converge as δ ↘ 0. In addition to λδ being
contact forms for δ>0, we also want the given open book decomposition to support
ker λδ, hence Xδ needs to be transverse to the pages of the open book decomposition
which is equivalent to γ ′ 1 (r) = 0. A curve γ (r) fulfilling these conditions can clearly
be constructed.
The following result shows that we can always assume that a Giroux contact form is
equal to any of the forms provided by Proposition 2.4.
PROPOSITION 2.5
Let M be a closed 3-dimensional manifold with contact structure ξ. Then, for every
δ>0, there is a diffeomorphism ϕδ : M → M such that ker λδ = ϕ∗ξ where λδ is
given by Proposition 2.4.
Proof
Existence of an open book decomposition supporting ξ follows from the existence
part of Giroux’s theorem. On the other hand, Proposition 2.4 yields contact forms λδ
such that ker λδ is also supported by the same open book decomposition as ξ for any
δ>0. By the uniqueness part of Giroux’s theorem, ξ and ker λδ are diffeomorphic.
It follows from our previous construction of the forms λδ that λ0 satisfies λ0 ∧dλ0 > 0
on ∂W ×D1−ε0 and that λ0 = dφ otherwise. For δ → 0, the Reeb vector fields Xδ will
converge to some vector field X0, which is the Reeb vector field of λ0 if r

44 CASIM ABBAS
the standard complex structure on the cylinder [0, +∞) × S 1 after introducing polar
coordinates near the punctures. Moreover, all the punctures are positive ∗ the family
(ũα)0≤α≤2π is a finite energy foliation, and the curves ũα are ˜Jδ-holomorphic near the
punctures.
Proof
We parameterize the leaves of the open book decomposition uα : ˙S → M, 0 ≤ α<
2π, and we assume that they look as follows near the binding:
uα :[0, +∞) × S 1 −→ S 1 × D1,
uα(s, t) = t,r(s)e iα ,
(2.6)
where r are smooth functions with lims→∞ r(s) = 0 to be determined shortly. We
use the notation (r, φ) for polar coordinates on the disk D = D1. We identify some
neighborhood U of the punctures of ˙S with a finite disjoint union of half-cylinders
[0, +∞) × S1 . Recall that the binding orbit is given by
t
x(t) = , 0, 0 , 0 ≤ t ≤ 2πγ1(0)
γ1(0)
and that it has minimal period T = 2πγ1(0). We define smooth functions aα : ˙S → R
by
s
aα(z) := 0 γ1
′ ′ r(s ) ds if z = (s, t) ∈ [0, +∞) × S1 ⊂ U,
0 if z/∈ U
so that
u ∗
α λ0 ◦ j = daα,
where j is a complex structure on ˙S which equals the standard structure i on [0, +∞)×
S 1 (i.e., near the punctures). We want to turn the maps ũα = (aα,uα) : ˙S → R × M
into ˜J0-holomorphic curves for a suitable almost-complex structure ˜J0 on R × M.
Recall that the contact structure is given by
∂ ∂
ker λδ = Span{η1,η2} = Span , −γ2(r)
∂r ∂θ
We define complex structures Jδ :kerλδ → ker λδ by
Jδ(θ,r,φ) − γ2(r) ∂
∂θ
∂
+ γ1(r) := −
∂φ
1
h(r)
∗ A puncture pj is called positive for the curve (aα,uα) if limz→pj aα(z) =+∞.
∂
+ γ1(r) .
∂φ
∂
∂r
(2.7)

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 45
and
Jδ(θ,r,φ) ∂
∂r
:= h(r) − γ2(r) ∂
∂θ
∂
+ γ1(r) ,
∂φ
where h :(0, 1] → R\{0} are suitable smooth functions. Also recall that γ1,γ2
depend on δ away from the binding orbit. We want Jδ to be compatible with dλδ,that
is, we want
dλδ(η1,Jη1) = h(r) µ(r) > 0 and dλδ(η2,Jη2) = µ(r)
> 0
h(r)
so that h(r) > 0. We also demand that Jδ extends smoothly over the binding {r = 0}.
Expressing the vectors η1 and η2 in Cartesian coordinates, we have
and
η1 = 1
r
x ∂ ∂
+ y
∂x ∂y
η2 =−γ2(r) ∂
∂θ + γ1(r) x ∂
∂y − γ1(r) y ∂
∂x .
We introduce the following generators of the contact structure:
and
We compute from this
Now
ε1 := γ1(r) ∂
∂y
= yγ1(r)
r
ε2 := γ1(r) ∂
∂x
= xγ1(r)
r
− xγ2(r)
r 2
η1 + x
η2
r2
∂
∂θ
yγ2(r)
+
r2 ∂
∂θ
η1 − y
η2.
r2 η1 = 1
rγ1(r) (yε1 + xε2), η2 = xε1 − yε2.
Jδε1 = yγ1(r)h(r)
η2 −
r
x
r2h(r) η1
1
=
r xyγ1(r)h(r)
xy
−
r3
ε1
h(r)γ1(r)
1
−
r y2 x
γ1(r)h(r) +
2
r3
γ1(r)h(r)
ε2

46 CASIM ABBAS
and
Jδε2 = xγ1(r)h(r)
η2 +
r
y
r2h(r) η1
= − 1
r xyγ1(r)h(r)
xy
+
r3
ε2
h(r)γ1(r)
1
+
r x2 y
γ1(r)h(r) +
2
r3
ε1.
γ1(r)h(r)
Inserting x = r cos φ, y = r sin φ, and demanding that the limit is φ-independent as
r → 0, we arrive at the condition that rh(r) γ1(r) ≡±1 for small r. Recalling that
we need h>0, we obtain
h(r) = 1
rγ1(r)
for small r.
As usual, we continue Jδ to an almost-complex structure ˜Jδ on R × M by setting
˜Jδ(θ,r,φ) ∂
∂τ
:= Xδ(θ,r,φ),
where τ denotes the coordinate in the R-direction. We emphasize that ˜Jδ also makes
sense for δ = 0. We now arrange r(s) in (2.6) such that the Giroux leaves ũα = (aα,uα)
become ˜J0-holomorphic curves. ∗ We compute for r ≤ 1 − ε0
∂sũα + ˜J0(uα)∂tũα = γ1(r) ∂ ∂
+ r′
∂τ ∂r +
˜J0(uα)
∂
∂θ
= γ1(r) ∂ ∂
+ r′
∂τ ∂r + ˜J0(uα) γ1(r) Xδ(uα)
+ ˜J0(uα)
∂
− γ1(r)Xδ(uα)
∂θ
′ ∂
= r
∂r + ′
˜J0(uα)
γ 1 (r)
γ1(r)
µ(r)
∂ ∂
− γ2(r)
∂φ ∂θ
= r ′ − γ ′ 1 (r)
∂
µ(r)h(r) ∂r ,
hence the Giroux leaves satisfy the equation if we choose r to be a solution of the
ordinary differential equation
r ′ (s) =
γ ′ 1 (r(s))
µ(r(s)) h(r(s)) .
∗ The calculation shows that we can make them Jδ-holomorphic for all δ ≥ 0 near the binding.

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 47
Note that r ′ (s) < 0. We also choose h(r) ≡ 1 for r ≥ 1 − ε0. We continue the
almost-complex structures Jδ :kerλδ → ker λδ (which were only defined near the
binding) smoothly to all of M. Away from the binding we have Xδ = ∂/∂φ, andwe
extend Jδ as before to T (R × M). Away from the binding, if δ = 0, wehavethat
ker λ0 coincides with the tangent spaces of the pages of the open book decomposition.
Because aα is constant away from the binding, the solutions ũα which we constructed
near the binding fit together smoothly with the pages of the open book decomposition
and solve the holomorphic curve equation for the almost-complex structure J0.
Remark 2.7
Near the binding orbit the function r(s) satisfies a differential equation of the form
and
r ′ (s) = r(s) r(s) := γ ′ 1 (r(s))γ1(r(s))
µ(r(s))
lim
r→0
(r) = γ ′′
1 (0)
γ ′′
2
(0) =: κ.
r(s),
Writing r(s) = c(s)e κs , the function c(s) satisfies c ′ (s) = ((r(s)) − κ)c(s), and
hence it is a decreasing function which converges to a constant as s →+∞.
We return to Examples 2.2 and 2.3, and we compute r(s) for large s. The differential
equation in the case of Example 2.3 for large s is
so that
r ′ (s) = γ ′ 1 (r(s))γ1(r(s))
r(s) =−kT
µ(r(s))
1 − r 2 (s) r(s),
r(s) =
1
√ 1 + ce 2kT s ,
where c is a constant. In Example 2.2, the differential equation reads
and solutions satisfy
r ′ (s) = γ ′ 1 (r(s))γ1(r(s))
r(s) =−
µ(r(s))
kT
1 − r2 (s) r(s),
r(s) = ce −kT s e (1/2)r2 (s) .
2.2. Functional analytic setup and the implicit function theorem
In the following theorem we prove the existence of a smooth family of solutions near a
given solution. In Proposition 2.6, we constructed a finite energy foliation for the data

48 CASIM ABBAS
(λ0,J0) with vanishing harmonic form. The form λ0, however, is only a confoliation
form. We produce solutions for the perturbed data (λδ,Jδ), and harmonic forms appear
if the surface S is not a sphere. The key result is an application of the implicit function
theorem in a suitable setting.
THEOREM 2.8
Assume one of the following.
(1) Let (a0,u0) : ˙S → R × M be one of the ˜J0-holomorphic curves described in
Proposition 2.6 with complex structure j0 on S (we refer to such u0 as a Giroux
leaf) and confoliation form λ0.
(2) Let ( ˙S,j0,a0,u0,γ0) be a solution of the differential equation (1.1) for some
dλ0-compatible complex structure J0 :kerλ0 → ker λ0 which, near the binding
orbit, agrees with (2.7), and where λ0 is a contact form which is a local
model near the binding. Assume that u0 is an embedding and that it is of the
form u0 = φg(v0), whereg : S → R is a smooth function, φ is the flow of the
Reeb vector field, and where v0 : ˙S → M is a Giroux leaf as in Proposition
2.6.
(3) Let Jδ be a smooth family of dλδ-compatible complex structures also agreeing
with (2.7) near the binding orbit, where (λδ)−ε

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 49
Note that this is well defined because u is transverse to the Reeb vector field so that
πTu(z) :TzS → ker λ(u(z)) is an isomorphism. By the second equation of (1.1), we
then have to solve the equation df ◦ jf + u ∗ 0 λ ◦ jf = da + γ for a,f,γ on ˙S which
is equivalent to the equation
¯∂jf (a + if ) = u ∗
0 λ ◦ jf − i(u ∗
0 λ) − γ − i(γ ◦ jf ). (2.9)
Recall that we are looking for a of the form a = a0 + b, where b is a suitable realvalued
function defined on the whole surface S. We obtain the differential equation
¯∂jf (b + if ) = u ∗
0 λ ◦ jf − i(u ∗
0 λ) − ¯∂jf a0 − γ − i(γ ◦ jf ), (2.10)
and it follows from a straightforward calculation (see the appendix) that all expressions
on the right-hand side of (2.10) are bounded near the punctures; in particular, they
are contained in the spaces L p (T ∗ S ⊗ C) for any p. This is what the assumption
κ ≤−(1/2) from Definition 2.1 is needed for. We work in the function space b +if ∈
W 1,p (S,C), where p>2. For any complex structure j on S, the space L p (T ∗ S ⊗ C)
of complex-valued 1-forms of class L p decomposes into complex linear and complex
antilinear forms (with respect to j). We use the notation
L p (T ∗ S ⊗ C) = L p (T ∗ S ⊗ C) 1,0
j ⊕ Lp (T ∗ S ⊗ C) 0,1
j .
The operator b + if ↦→ ¯∂jf (b + if ) is then a section in the vector bundle
L p (T ∗ S ⊗ C) 0,1 :=
b+if ∈W 1,p (S,C)
{b + if }×L p (T ∗ S ⊗ C) 0,1
jf → W 1,p (S,C).
This vector bundle is of course trivial, but here are some explicit local trivializations
for f, g ∈ W 1,p (S,R) sufficiently close to each other:
If we write
we see that fg is invertible with
fg : Lp (T ∗S ⊗ C) 0,1
jf −→Lp (T ∗S ⊗ C) 0,1
jg
τ ↦−→ τ + i(τ ◦ jg).
τ + i(τ ◦ jg) = τ ◦ (IdTS − jf ◦ jg),
−1
fg τ = τ ◦ (IdTS − jf ◦ jg) −1 .
(2.11)
It follows from the Hodge decomposition theorem that every cohomology class [σ ] ∈
H 1 (S,R) has a unique harmonic representative ψj(σ ) ∈ H 1 j (S), where H 1 j (S) is

50 CASIM ABBAS
defined as
H 1
j (S) := γ ∈ E 1 (S) dγ = 0, d(γ ◦ j) = 0
(2.12)
and where E 1 (S) denotes the space of all (smooth) real-valued 1-forms on S, and
we write E 0,1 (S) = E 0,1
j (S) for the space of complex antilinear 1-forms on S with
respect to j, that is, complex-valued 1-forms σ such that iσ + σj = 0. Note that
our definition coincides with the set of closed and co-closed 1-forms on S. Moreover,
by elliptic regularity, we may also consider Sobolev forms. We identify H 1 (S,R)
with R 2g , and we consider the following parameter-dependent section in the bundle
L p (T ∗ S ⊗ C) 0,1 → W 1,p (S,C)
F : W 1,p (S,C) × R 2g −→ L p (T ∗ S ⊗ C) 0,1
F (b + if, σ ):= ¯∂jf (b + if ) − u ∗
0λ ◦ jf + i(u ∗
0λ) + ¯∂jf a0 + ψjf (σ ) + i
ψjf (σ ) ◦ jf
(2.13)
with jf as in (2.8). Recalling that z ↦→ jf (z) may not be differentiable, we interpret
the equation d(γ ◦ jf ) = 0 in the sense of weak derivatives. The solution set of (2.10)
is then the zero set of F . We consider the real parameter δ which we dropped from
the notation, fixed at the moment. For g ≡ 0 and b + if small in the W 1,p -norm,
we consider the composition ˆF (b + if, σ ) = fg(F (b + if, σ )). Its linearization
in the point (b + if, σ ) = (0,σ0), where σ0 is defined by ψj0(σ0) = γ0 and where
F (0,σ0) = 0,is
where
D ˆF (0,σ0) :W 1,p (S,C) × R 2g −→ L p (T ∗ S ⊗ C) 0,1
j0
D ˆF (0,σ0)(ζ,σ) = ¯∂j0ζ + ψj0(σ ) + i(ψj0(σ ) ◦ j0) + Lζ,
L : W 1,p (S,C) → W 1,p (T ∗ S ⊗ C) 0,1
j0 ↩→ L p (T ∗ S ⊗ C) 0,1
j0
is the compact linear map
where
Lζ =− 1
2 u∗
0 λ ◦ (Aζ + j0Aζj0) + i
2 u∗
0 λ ◦ (j0Aζ − Aζj0) + Bζ + iBζj0,
Bζ = d
dτ
τ=0
ψjτ k(σ0), ζ = h + ik,

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 51
and where
Aζ = d
jτ
dτ k = h (πλTu0)
τ=0
−1 [J (u0)DXλ(u0)
− DXλ(u0)J (u0) + DJ(u0)Xλ(u0)](πλTu0).
The linear term L therefore does not contribute to the Fredholm index of D ˆF (0,σ0).
We note that the linear map ζ ↦→ Lζ only depends on the imaginary part of ζ .We
claim that the operator
W 1,p (S,C) × R 2g −→ L p (T ∗ S ⊗ C) 0,1
j0
(ζ,σ) ↦−→ ¯∂j0ζ + ψj0(σ ) + i(ψj0(σ ) ◦ j0)
is a surjective Fredholm operator of index 2. Then we would have ind(D ˆF (0,σ0)) = 2
as well. Here is the argument: The Riemann-Roch theorem asserts that the kernel and
the cokernel of the Cauchy-Riemann operator ¯∂j (acting on smooth complex-valued
functions on S) are both finite-dimensional and that
dimR ker ¯∂j
0,1
− dimR E (S)/Im ¯∂j = 2 − 2g,
where g is the genus of the surface S. The only holomorphic functions on S are the
constant functions, hence E0,1 (S)/Im ¯∂j has dimension 2g.
On the other hand, the vector space H 1 j (S) of all (real-valued) harmonic 1-forms
on S also has dimension 2g (see [15]). We now consider the linear map
: H 1
j (S) −→ E0,1 (S)/Im ¯∂j
(γ ):= [γ + i(γ ◦ j)],
where [ . ] denotes the equivalence classes of (0, 1)-forms. Assume that (γ ) = [0],
that is, that there is a complex-valued smooth function f = u+iv on S such that ¯∂jf =
γ +i(γ ◦j).Sinceγ is a harmonic 1-form, we conclude that d(dv◦j) = d(du◦j) = 0
(i.e., both u and v are harmonic). Since there are only constant harmonic functions
on S we obtain γ = 0 (i.e., is injective and also bijective). Hence, (0, 1)-forms
γ + i(γ ◦ j) with γ ∈ H 1 j (S) make up the cokernel of ¯∂j : C ∞ (S,C) → E 0,1 (S).
This proves the claim that the operator D ˆF (0,σ0) is Fredholm of index 2. We
now show that the operator D ˆF (0,σ0) is surjective. Using the decomposition
L p (T ∗ S ⊗ C) 0,1
j0 = R(¯∂j0) ⊕ H 1
j0 (S)

52 CASIM ABBAS
and denoting the corresponding projections by π1,π2, we see that it suffices to prove
the surjectivity of the operator
T : W 1,p (S,C) → R(¯∂j0)
Tζ := ¯∂j0ζ + π1(Lζ ),
which is a Fredholm operator of index 2. Assume that ζ ∈ ker T .Unlessζ ≡ 0, the
set {z ∈ S | ζ (z) = 0} consists of finitely many points by the similarity principle (see
[23]), and the local degree of each zero is positive. On the other hand, the sum of all
the local degrees has to be zero, hence elements in the kernel of T are nowhere zero.
Actually, if h + ik ∈ ker T ,thenevenk is nowhere zero because h + c + ik ∈ ker T
for any real constant c since the zero-order term L only depends on the imaginary part
of ζ . Therefore, ∗ dim ker T ≤ 2, and since the Fredholm index of T equals 2, we
actually have dim ker T = 2. This proves the surjectivity of T and also of D ˆF (0,σ0)
so that the set M of all pairs (b + if, γ ) solving the differential equation (2.10) isa
2-dimensional manifold with T(0,γ0)M = ker D ˆF (0,γ0). If we add a real constant to
b + if , then we obtain again a solution of (2.10). If we divide M by this R-action,
then we obtain a 1-dimensional family of solutions (ũτ )−ε

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 53
where the map r satisfies for every (i, j) ∈ N 2 a decay estimate of the form
with Mij and d positive constants.
|∇ i
s ∇j
t r(s, t)| ≤Mij e −ds
Our situation is less general than in [31], so we will explain the notation in the context
of this paper. The setup is a manifold M with contact form λ and contact structure
ξ = ker λ. Consider a periodic orbit ¯P of the Reeb vector field Xλ with period T ,and
we may assume here that T is its minimal period. We introduce P (t) := ¯P (Tt/2π)
such that P (0) = P (2π). IfJ : ξ → ξ is a dλ-compatible complex structure, then
the set of all ˜J -holomorphic half-cylinders
ũ = (a,u) :[R,∞) × S 1 → R × M, S 1 = R/2πZ
for which |a(s, t) − Ts/2π| and |u(s, t) − P (t)| decay at some exponential rate (in
local coordinates near the orbit P (S 1 )) is denoted by M(P,J). Note that it is assumed
here that the domain [R,∞) × S 1 is endowed with the standard complex structure. A
smooth map U :[R,∞)×S 1 → P ∗ ξ for which U(s, t) ∈ ξP (t) is called an asymptotic
representative of ũ if there is a proper embedding ψ :[R,∞) × S 1 → R × S 1
asymptotic to the identity so that
ũ ψ(s, t) = Ts/2π, exp P (t) U(s, t) , ∀ (s, t) ∈ [R,∞) × S 1
(exp is the exponential map corresponding to some metric on M, e.g., the one induced
by λ and J ). Every ũ ∈ M(P,J) has an asymptotic representative (see [31]). The
asymptotic operator AP,J is defined as follows:
(AP,Jh)(t) :=− T
2π J P (t) d
ds
s=0
Dφ−s(φs(P (t)))h(φs(P (t))) ,
where φs is the flow of the Reeb vector field and where h is a section in P ∗ ξ →
S 1 . Because the Reeb flow preserves the splitting TM = R Xλ ⊕ ξ we have also
(AP,Jh)(t) ∈ ξP (t).
We compute the asymptotic operator AP,J for the binding orbit
¯P (t) =
t
, 0, 0 ∈ S
γ1(0) 1 × R2 .
Recall that the above periodic orbit has minimal period T = 2πγ1(0). Using
φs P (t) = φs(t,0, 0) = t + s
, 0, 0 ,
γ1(0)

54 CASIM ABBAS
formula (2.2) for the linearization of the Reeb flow with h(t) = (0,ζ(t),η(t)), and
the fact that J (t,0, 0) ∂ ∂
∂
= and J (t,0, 0) ∂x ∂y ∂y =−∂ , we compute
∂x
′ h (t)
(AP,Jh)(t) =−γ1(0)J (t,0, 0)
γ1(0) +
0 β(0) ζ (t)
−β(0) 0 η(t)
where J0 =
if
0 −1
1 0
=−J0 h ′ (t) − γ1(0)β(0) h(t)
=−J0h ′ (t) + κh(t),
and κ = γ ′′ ′′
1 (0)/γ 2 (0) ∈ (−1, 0). Hence λ ∈ σ (AP,J) precisely
h ′ (t) = (λ − κ) J0 h(t) and h(2π) = h(0)
(i.e., σ (AP,J) ={κ + l | l ∈ Z}), and the largest negative eigenvalue is given by κ.
The corresponding eigenspace consists of all constant vectors h(t) ≡ const ∈ R 2 .
The eigenspace for the eigenvalues κ + l consists of all
h(t) = e J0lt h0, with h0 ∈ R 2 .
THEOREM 3.2 (Compactness)
Let λ be a contact form on M which is a local model near the binding (of the Giroux
leaf v0), and let J :kerλ→ker λ be a dλ-compatible complex structure. Consider a
smooth family of solutions (S,jτ ,aτ ,uτ ,γτ ,J)0≤τ

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 55
Remark 3.3
We may assume without loss of generality that v0 ≡ u0 and f0 ≡ 0.Ifz ∈ ˙S,thenwe
denote by T (z) > 0 the positive return time of the point u0(z); that is, we have
T (z) := inf T>0 φT (u0(z)) ∈ u0( ˙S) < +∞.
We claim that the return time z ↦→ T (z) extends continuously over the punctures of
the surface, and that therefore there is an upper bound
T := sup T (z) < ∞.
z∈ ˙S
Using (2.1)and(2.6), we note that, asymptotically near the punctures, φT (u0(s, t)) ∈
S 1 × R 2 has the following structure:
φT (u0(s, t)) = t + α(r(s))T, r(s) exp[i(α0 + β(r(s))T )] ,
where r(s) is a strictly decreasing function, α0 is some constant, and α(r),β(r) are
suitable functions for which the limits limr→0 β(r) and limr→0 α(r) exist and are not
zero. Hence, if T = T (u0(s, t)) is the positive return time at the point u0(s, t), then
T u0(s, t) =
and therefore the limit for s →+∞exists.
2π
|β(r(s))| ,
The remainder of this section is devoted to the proof of Theorem 3.2. We recall
that the functions aτ and fτ satisfy the Cauchy-Riemann type equation (2.9)whichis
¯∂jτ (aτ + ifτ ) = u ∗
0 λ ◦ jτ − i(u ∗
0 λ) − γτ − i(γτ ◦ jτ ),
where the complex structure jτ is given by (2.8)or
jτ (z) = πλTu0(z) −1 Tφfτ (z)(u0(z)) −1 · J φfτ (z)(u0(z))
Tφfτ (z) u0(z) πλTu0(z),
and that γτ is a closed 1-form on S with d(γτ ◦ jτ ) = 0.
The following L ∞ -bound is the crucial ingredient for the compactness result. We
claim that
sup
0≤τ

56 CASIM ABBAS
Restricting any of the solutions to a simply connected subset U ⊂ ˙S, we can
write γτ = dhτ for a suitable function hτ : U → R, and the maps
ũτ : U → R × M, ũτ = (aτ + hτ ,uτ )
are ˜J -holomorphic curves. If two such curves ũτ and ũτ ′ have an isolated intersection,
then the corresponding intersection number is positive (see [28], [5], or [27] for
positivity of (self-)intersections for holomorphic curves). We claim that
u0( ˙S) ∩ uτ ( ˙S) =∅, ∀ 0 0 is different from u0( ˙S), we conclude that if uτ and u0 intersect,
then the intersection point of the corresponding holomorphic curves ũτ and ũ0 must
be isolated. But on the other hand, this implies that uτ ′ and u0 would also intersect for
all τ ′ sufficiently close to τ by positivity of the intersection number showing that the
set O is open.
We conclude from the above that we have a sequence τk ↘ ˜τ and points pk,qk ∈ ˙S
such that uτk(pk) = u0(qk). Passing to a suitable subsequence, we may assume
convergence of the sequences (pk)k∈N and (qk)k∈N to points p, q ∈ S. Because of
u˜τ ( ˙S) ∩ u0( ˙S) =∅the points p, q must be punctures, and they have to be equal
z0 = p = q ∈ S\ ˙S. The reason for this is the following. The maps uτk,u0 are
asymptotic near the punctures to a disjoint union of finitely many periodic Reeb
orbits which are not iterates of other periodic orbits. Also, different punctures always
correspond to different periodic orbits. This follows from Giroux’s result and our
constructions in Section 2 of this paper.

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 57
We now derive a contradiction using Siefring’s result. The harmonic forms γτk in
equation (1.1) are defined on all of S. Hence they are exact on some open neighborhood
U of the puncture z0,andγτk = dhτk for suitable functions hτk on U and similarly γ˜τ =
dh˜τ . We may also assume that j˜τ |U = jτk|U = j0 after changing local coordinates
near z0. Then on the set U, the maps ũτk = (aτk + hτk,uτk) and ũ0 = (a0 + h0,u0) are
holomorphic curves with ũτk(pk) = ũ0(qk) while the images of ũ˜τ and ũ0 have empty
intersection. Now let
U˜τ ,Uτk, U0 :[R,∞) × S 1 → R 2
be asymptotic representatives of the holomorphic curves ũ˜τ , ũτk, ũ0, respectively.
Invoking Theorem 3.1 and our subsequent computation of the asymptotic operator
and its spectrum, we obtain the following asymptotic formulas
Uτ (s, t) − U0(s, t) = e λτ s eτ (t) + rτ (s, t) , τ = ˜τ,τk, s ≥ Rτ , (3.2)
where Rτ > 0 is some constant and where λτ < 0 is some negative eigenvalue of
the asymptotic operator AP,J. It is of the form λτ = κ + lτ , where lτ is an integer,
κ = γ ′′
1
(0)/γ ′′
2 (0) is not an integer, and where eτ (t) = e J0lτ t hτ , hτ ∈ R 2 \{0} is an
eigenvector corresponding to the eigenvalue λτ = κ + lτ . Note that the above formula
applies since Uτ − U0 cannot vanish identically. We will actually show that lτ ≡ 0.
The asymptotic representative U0 is given by
u0(s, t) = t,r(s)e iα0
= t,U0(s, t) ,
using equation (2.6), and we recall that r(s) = c(s)eκs , where c(s) → c∞ > 0 as
s →+∞. An asymptotic representative of ũτ , however, is given by an expression
such as
uτ ψ(s, t) = t,Uτ (s, t) ,
where ψ :[R,∞) × S 1 → R × S 1 is a proper embedding converging to the identity
map as s →+∞. Writing (s ′ ,t ′ ) = ψ(s, t), we get using equations (2.1) forthe
Reeb flow
Uτ (s, t) = c(s ′ )e κs′
e i(α0+β(r(s ′ ))fτ (s ′ ,t ′ ))
= e κs eτ + rτ (s, t) .
The asymptotic formula for Uτ a priori allows for other decay rates, but κ is the only
possible one. Dividing by e κs and passing to the limit s →+∞, we obtain
eτ = c∞e iα0 e iβ(0)fτ (∞) ,

58 CASIM ABBAS
where fτ (∞) = lims→+∞ fτ (s, t) which is independent of t since fτ extends continuously
over the punctures. Hence the difference Uτ − U0 has decay rate λτ ≡ κ as
claimed unless the two eigenvectors eτ and e0 agree, which is equivalent to
fτ (∞) ∈ 2π
β(0) Z
or τ = ˜τ in our case. The maps Uτ − U0 satisfy a Cauchy-Riemann type equation
to which the similarity principle applies so that, for every zero (s, t) of Uτ − U0, the
map σ ↦→ (Uτ − U0)(s + ɛ cos σ, t + ɛ sin σ ) has positive degree for small ɛ>0.
The Cauchy-Riemann type equation mentioned above is derived in [31, Section 5.3]
as well as in [1, Section 3] in a slightly different context, and also in [21]. If R is
sufficiently large, then the map
S 1 → S 1 , t ↦→ Wτ (R,t) := Uτ − U0
|Uτ − U0| (R,t)
is well defined, and it has degree lτ because the remainder term rτ (s, t) decays
exponentially in s. Zeros of Uτ − U0 contribute in the following way: if R ′ 0 so large that
degW˜τ (R ′ , ·) = l˜τ < 0.Forτ sufficiently close to ˜τ,wealsohavedegWτ (R ′ , ·) = l˜τ .
On the other hand, we have degWτ (R,·) = 0 for R>R ′ sufficiently large. Equation
(3.3) implies that the map Uτ − U0 must have zeros in [R ′ ,R] × S 1 to account
for the difference in degrees, but we know that there are none for τ < ˜τ. This
contradiction shows that l˜τ = 0 is impossible. Choose again R ′ > 0 so large that
degW˜τ (R ′ , ·) = 0. The degree does not change if we slightly alter τ. In particular,
we have degWτ (R ′ , ·) = 0 for τ > ˜τ close to ˜τ as well. For R>>R ′ ,wehave
degW˜τ (R,·) = 0, and we recall that
(Uτk − U0)(sk,tk) = 0, τk ↘ ˜τ
for a suitable sequence (sk,tk) with sk →+∞and that the set of zeros of Uτk − U0 is
discrete. This, however, contradicts equation (3.3) since the zeros have positive orders.
Summarizing, we have shown that the assumption O = ∅leads to a contradiction
which implies the a priori bound (3.1).
The monotonicity of the functions fτ in τ and the bound (3.1) imply that the
functions fτ converge pointwise to a measurable function fτ0 as τ ↗ τ0. Wealso

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 59
know that fτ0 L ∞ ( ˙S) ≤ T . We then obtain a complex structure jτ0 on ˙S by
jτ0(z) = πλTu0(z) −1 Tφfτ (z)(u0(z)) 0 −1 × J φfτ (z)(u0(z)) 0
Tφfτ (z) u0(z) 0 πλTu0(z).
By definition, the complex structure jτ0 is also of class L∞ and jτ (z) → j1(z)
pointwise. Our task is to improve the regularity of the limit fτ0 and the character of
the convergence fτ → fτ0 . We also have to establish convergence of the functions aτ
for τ ↗ τ0. The complex structures jτ are of course all smooth, but the limit jτ0 might
only be measurable.
3.1. The Beltrami equation
For the reader’s convenience, we briefly summarize a few classical facts from the
theory of quasiconformal mappings (see [8], [9]). The punctured surface ˙S carries
metrics gτ , also of class L ∞ for τ = τ0 and smooth otherwise, so that
In fact, gτ is given by
gτ (z) jτ (z)v, jτ (z)w = gτ (z)(v, w), for all v, w ∈ Tz ˙S.
gτ (z)(v, w) = dλ uτ (z) πλTuτ (z)v, J(uτ (z))πλTuτ (z)w .
In the case τ = τ0, we replace πλTuτ (z) by Tφfτ 0 (z)(u0(z))πλTu0(z). Wehave
sup τ gτ L ∞ ( ˙S) < ∞ and gτ → gτ0 pointwise as τ ↗ τ0. Our considerations about
the regularity of the limit are of local nature, so we may replace ˙S with a ball B ⊂ C
centered at the origin. Denoting the metric tensor of gτ by (g τ kl )1≤k,l≤2, wedefinethe
following complex-valued smooth functions:
and we note that
µτ (z) :=
1
2 (gτ 11 (z) − gτ 22 (z)) + igτ 12 (z)
1
2 (gτ 11 (z) + gτ 22 (z)) + gτ 11 (z)gτ 22 (z) − (gτ ,
12 (z))2
sup µτ L∞ ( ˙S) < 1
τ
and that µτ → µτ0 pointwise. We view the functions µτ as functions on the whole
complex plane by trivially extending them beyond B. Then they are also τ-uniformly
bounded in L p (C) for all 1 ≤ p ≤∞and µτ → µτ0 in Lp (C) for 1 ≤ p

60 CASIM ABBAS
for τ

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 61
We used here the notation ∂ = (1/2)(∂s + i∂t) and ∂ = (1/2)(∂s − i∂t). Properties
(1) and (2) above should be understood in the sense of distributions. The proof of (4)
involves the Calderón-Zygmund inequality and the Riesz-Thorin convexity theorem
(see [25] and[32]). Following [9], we define Bp with p>2 to be the space of all
locally integrable functions on the plane which have weak derivatives in Lp (C),vanish
in the origin, and which satisfy a global Hölder condition with exponent 1 − 2
p .For
u ∈ Bp, we then define a norm by
|u(z2) − u(z1)|
uBp := sup
z1=z2 |z2 − z1| 1−(2/p) +∂uLp (C) +∂uLp (C)
so that Bp becomes a Banach space. We usually choose p > 2 such that
cp sup τ µτ L ∞ (C) < 1, where cp is the constant from item (4) above.
THEOREM 3.4 ([9, Theorem 1])
Assume that p>2 such that cp sup τ µτ L ∞ (C) < 1.Ifσ ∈ L p (C), then the equation
has a unique solution u = uµ,σ ∈ Bp.
∂u = µ∂u+ σ
For the existence part of the theorem, one first solves the following fixed-point problem
in L p (C):
This is possible because the map
q = Ɣ(µq) + Ɣσ.
L p (C) −→ L p (C)
q ↦−→ Ɣ(µq + σ )
is a contraction in view of cpµL ∞ (C) < 1. Then
u := A(µq + σ )
is the desired solution. The following estimate is also derived in [9]:
qL p (C) ≤ c ′
p σ L p (C)
with c ′ p = cp/(1 − cpµL ∞ (C)), which follows from
qL p (C) ≤Ɣ(µq)L p (C) +ƔσL p (C)
≤ cpµL ∞ (C)qL p (C) + cpσ L p (C).
(3.4)

62 CASIM ABBAS
Recalling our original situation, we have the following result which shows that
there is some sort of conformal mapping for j1 on the ball B.
THEOREM 3.5 ([9, Theorem 4])
Let µ : C → C be an essentially bounded measurable function with µ|C\B ≡ 0 and
p>2 such that cpµL ∞ (C) < 1. Then there is a unique map α : C → C with
α(0) = 0 such that
∂α = µ∂α
in the sense of distributions with ∂α − 1 ∈ L p (C).
The desired map α is given by
α(z) = z + u(z),
where u ∈ Bp solves the equation ∂u = µ∂u + µ. In particular, α ∈ W 1,p (B).
Lemma 8 in [9] states that α : C → C is a homeomorphism. We can apply the
theorem to all the µτ , 0 2 such that cp sup n µnL ∞ (C) < 1 and any compact set
B ⊂ C.
Proof
We first estimate with g ∈ Lp (C) and z = 0
|Ag(z)| = 1
z
g(ξ) dξ d¯ξ
2π C ξ(ξ − z)
≤ |z|
2π gLp
1
(C)
ξ(ξ − z)
≤ CpgLp 2
1−
(C) |z| p ,
L p
p−1 (C)
(3.5)

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 63
where the last estimate holds in view of
|ξ(ξ − z)| −p/(p−1) dξ d¯ξ ζ =z−1
ξ
=
C
If q solves q = Ɣ(µq + µ),then
C
=|z| 2−2p/(p−1)
¯∂(αn − α) = µn ∂(αn − α) − µ∂α+ µn ∂α
|z 2 ζ 2 − z 2 ζ | −p/(p−1) |z| 2 dζ d¯ζ
|ζ (ζ − 1)| −p/(p−1) dζ d¯ζ .
C
2πCp
= µn ∂(αn − α) + µn − µ + (µn − µ)Ɣ(µq + µ),
that is, the difference αn − α again satisfies an inhomogeneous Beltrami equation. By
Theorem 3.4, wehave
αn − α = A(µnqn + λn),
where λn = µn − µ + (µn − µ)Ɣ(µq + µ) and where qn ∈ L p (C) solves qn =
Ɣ(µnqn + λn). Combining this with (3.5)and(3.4), we obtain
|αn(z) − α(z)| ≤Cp µnqn + λnL p (C) |z| 1−2/p
≤ (Cp sup µnL
n
∞ (C) · c ′
pλnLp (C) + Cp λnLp (C)) |z| 1−2/p . (3.6)
Since µn − µLp (C) → 0 and (µn − µ)Ɣ(µq + µ)Lp (C) → 0 by Lebesgue’s
theorem we also have λnLp (C) → 0 and therefore αn → α uniformly on compact
sets. Since ¯∂(αn − α) = µn ∂(αn − α) + λn and αn − α = A(µnqn + λn), we verify
that
and
∂(αn − α) = Ɣ(µnqn + λn) = qn
¯∂(αn − α) = µnqn + λn.
Invoking (3.4) once again, we see that both ∂(αn − α)L p (C) and ¯∂(αn − α)L p (C)
can be bounded from above by a constant times λnL p (C) which converges to
zero.
We also need some facts concerning the classical case where µ ∈ C k,α (BR(0)),
BR(0) ={z ∈ C ||z|

64 CASIM ABBAS
THEOREM 3.7
Let µ, γ, δ ∈ C k,α (BR ′(0)) with 0

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 65
for a suitable constant c>0 depending on α and R. This is only implicitly proved in
[11], so we sketch the proof of this inequality. We have
and
(T2 − T1)h(z) = A (µ2 − µ1)∂h + (γ2 − γ1)h (z)
− zƔ (µ2 − µ1)∂h + (γ2 − γ1)h (0),
∂ (T2 − T1)h (z) = Ɣ (µ2 − µ1)∂h + (γ2 − γ1)h (z)
− Ɣ (µ2 − µ1)∂h + (γ2 − γ1)h (0),
¯∂ (T2 − T1)h (z) = (µ2 − µ1)(z)∂h(z) + (γ2 − γ1)(z)h(z).
We need [13, (21)–(24)]. Adapted to our notation, they look as follows with z, z1,z2 ∈
BR(0):
Recalling that
and
|(Ah)(z2) − (Ah)(z1)|
|z2 − z1| α
|(Ah)(z)| ≤4RhC 0 (BR(0))
|(Ɣh)(z)| ≤ 2α+1
α Rα hC 0,α (BR(0))
|(Ɣh)(z2) − (Ɣh)(z1)|
|z2 − z1| α
≤ Cα hC0,α (BR(0)).
≤ 2hC 0 (BR(0)) + 2α+2
α Rα hC 0,α (BR(0))
hC 1,α (BR(0)) := hC 0 (BR(0)) +∂hC 0,α (BR(0)) +¯∂hC 0,α (BR(0))
and that the Hölder norm satisfies
kC 0,α (BR(0)) := kC0 |k(z2) − k(z1)|
(BR(0)) + sup
z1=z2 |z2 − z1| α
hkC 0,α (BR(0)) ≤ C hC 0,α (BR(0)) kC 0,α (BR(0))
for a suitable constant C depending only on α and R, the asserted inequality for the
operator norm of T2 − T1 follows. In the same way, we obtain
g2 − g1C 1,α (BR(0)) ≤ c δ2 − δ1C 0,α (BR(0)).

66 CASIM ABBAS
Since
w2 − w1C 1,α (BR(0)) ≤(T2 − T1)w2C 1,α (BR(0))
+ θ w2 − w1C 1,α (BR(0)) +g2 − g1C 1,α (BR(0))
and since θ

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 67
The integral
d(fγ) vanishes by Stokes’s theorem since fγ is a smooth 1-form on
S
the closed surface S. The form da∧γ ◦jf is not smooth on S, but the integral vanishes
anyway for the following reason. As we have proved in the appendix, the form γ ◦ jf
is bounded near the punctures, and hence in local coordinates near a puncture it is of
the form
σ = F (w1,w2)dw1 + F2(w1,w2)dw2, w1 + iw2 ∈ C,
where F1,F2 are smooth (except possibly at the origin) but bounded. Passing to polar
coordinates via
φ :[0, ∞) × S 1 −→ C\{0}
φ(s, t) = e −(s+it) = w1 + iw2,
we see that φ ∗ σ has to decay at the rate e −s for large s. The form da has γ1(r(s)) ds
as its leading term. Computing the integral
Ɣ a(γ ◦ jf ) over small loops Ɣ around
the punctures and using Stokes’s theorem, we conclude that the contribution from
neighborhoods
of the punctures can be made arbitrarily small. Therefore, the integral
˙S da ∧ γ ◦ jf must vanish.
If is a volume form on S, then we may write u∗ 0λ ∧ γ = g · for a suitable
smooth function g. Defining
|u
˙S
∗
0λ ∧ γ | := |g| ,
˙S
we have
γ 2
L2 ,jf =
u
˙S
∗
0λ ∧ γ
≤ |u ∗
0λ ∧ γ |
˙S
≤u ∗
0 λL 2 ,jf γ L 2 ,jf ,
which implies the assertion.
We resume the proof of the compactness result, Theorem 3.2. All the considerations
which follow are local. The task is to improve the regularity of the limit fτ0 and the
nature of the convergence fτ → fτ0 . Because the proof is somewhat lengthy, we
organize it in several steps. For τ

68 CASIM ABBAS
be the conformal transformations as in Section 2, that is,
Tατ (z) ◦ i = jτ (ατ (z)) ◦ Tατ (z), z ∈ B.
The L ∞ -bound (3.1) on the family of functions (fτ ) and the above L 2 -bound imply
convergence of the harmonic forms α ∗ τ γτ after maybe passing to a subsequence.
PROPOSITION 3.9
Let τ ′ k be a sequence converging to τ0, and let B ′ = Bε ′(0) with B′ ⊂ B. Then there is
a subsequence (τk) ⊂ (τ ′ k ) such that the harmonic 1-forms α∗ τk γτk converge in C∞ (B ′ ).
Proof
First, the harmonic 1-forms α ∗ τ γτ satisfy the same L 2 -bound as in Proposition 3.8:
α ∗
τ γτ 2
L2 (B) =
=
=
B
B
Uτ
α ∗
τ γτ ◦ i ∧ α ∗
τ γτ
α ∗
τ (γτ ◦ jτ ) ∧ α ∗
τ γτ
γτ ◦ jτ ∧ γτ
≤u ∗
0 λL 2 ,jτ
≤ C,
where C is a constant depending only on λ and u0 since
We write
sup jτ L∞ ( ˙S) < ∞.
τ
α ∗
τ γτ = h 1
τ ds + h2
τ dt,
where hk τ , k = 1, 2 are harmonic and bounded in L2 (B) independent of τ. Ify ∈ B
and BR(y) ⊂ BR(y) ⊂ B, then the classical mean-value theorem
h k 1
τ (y) =
πR2
h k
τ (x)dx
implies that, for any ball Bδ = Bδ(y) with Bδ ⊂ Bδ ⊂ B, we have the rather generous
estimate
h k
τ C0 (Bδ(y)) ≤ 1
√ h
πδ k
τ L2 √
C
(B) ≤ √ .
πδ
BR(y)

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 69
With y ∈ B and with ν = (ν1,ν2) being the unit outer normal to ∂Bδ(y), weget
∂sh k 1
τ (y) =
πδ2
∂sh
Bδ(y)
k
τ (x)dx
= 1
πδ2
div(h
Bδ(y)
k
τ , 0) dx
= 1
πδ2
h k
τ ν1 ds
and
|∇h k 1
τ (y)| =
πδ 2
∂Bδ(y)
h
∂Bδ(y)
k
τ νds
≤ 2
δ hk
τ C 0 (Bδ(y))
so that, for B ′ = Bε ′, r being the radius of B, andδ = r − ε′ ,wehave
∇h k
τ C 0 (B ′ ) ≤ 2√ C
√ πδ 2 .
By iterating this procedure on nested balls we obtain τ-uniform C 0 (B ′ )-bounds on all
derivatives. Convergence as stated in the proposition then follows from the Ascoli-
Arzela theorem.
3.3. A uniform L p -bound for the gradient
The first step of the regularity story is showing that the gradients of aτ + ifτ are
uniformly bounded in L p (B ′ ) for some p>2 and for any ball B ′ with B ′ ⊂ B. It will
soon become apparent why this gradient bound is necessary. Since we do not have a
lot to start with, the proof will be indirect. Recall the differential equation (2.9)
where
We set
¯∂jτ (aτ + ifτ ) = u ∗
0 λ ◦ jτ − i(u ∗
0 λ) − γτ − i(γτ ◦ jτ ),
jτ (z) = πλTu0(z) −1 Tφfτ (z)(u0(z)) −1 ×J φfτ (z)(u0(z))
Tφfτ (z) u0(z) πλTu0(z).
φτ (z) := aτ (z) + ifτ (z), z ∈ Uτ

70 CASIM ABBAS
so that, for z ∈ B, wehave
∂(φτ ◦ ατ )(z) = ∂jτ φτ ατ (z) ◦ ∂sατ (z)
= u ∗
0λ ◦ jτ − i(u ∗
0λ)ατ (z) ◦ ∂sατ (z)
− (α ∗
τ γτ )(z) · ∂
+ i(α∗ τ
∂s γτ )(z) · ∂
∂t
and
=: ˆFτ (z) + ˆGτ (z)
=: ˆHτ (z),
(3.8)
sup ˆFτ L
τ
p (B) < ∞ for some p>2 (3.9)
since ατ → ατ0 in W 1,p (B) and supτ jτ L∞ < ∞. Wealsohave
sup ˆGτ C
τ
k (B ′ ) < ∞ (3.10)
for any ball B ′ ⊂ B ′ ⊂ B and any integer k ≥ 0 in view of Proposition 3.9 (the
proposition asserts uniform convergence after passing to a suitable subsequence, but
uniform bounds on all derivatives are established in the proof). We claim now that,
for every ball B ′ ⊂ B ′ ⊂ B, there is a constant CB ′ > 0 such that
∇(φτ ◦ ατ )L p (B ′ ) ≤ CB ′, ∀ τ ∈ [0,τ0). (3.11)
Arguing indirectly, we may assume that there is a sequence τk ↗ τ0 such that
Now define
∇(φτk ◦ ατk)L p (B ′ ) →∞ for some ball B ′ ⊂ B ′ ⊂ B. (3.12)
εk := inf ε>0 ∃ x ∈ B ′ : ∇(φτk ◦ ατk)L p (Bε(x)) ≥ ε 2/p−1 ,
which are positive numbers since ε (2/p)−1 →+∞. Because we assumed (3.12) we
must have infk εk = 0, hence we will assume that εk → 0. Otherwise, if ε0 =
(1/2) infk εk > 0, then we cover B ′ with finitely many balls of radius ε0, andwe
would get a k-uniform L p -bound on each of them, contradicting (3.12). We claim that
∇(φτk ◦ ατk)L p (Bε k (x)) ≤ ε (2/p)−1
k , ∀ x ∈ B ′ . (3.13)
Otherwise, we could find y ∈ B ′ so that
∇(φτk ◦ ατk)Lp (Bε (y)) >ε k (2/p)−1
k ,

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 71
and we would still have the same inequality for a slightly smaller ε ′ k 0, and we define for z ∈ BR(0) the
functions
ξk(z) := (φτk ◦ ατk) xk + εk(z − xk) ,
which makes sense if k is sufficiently large. The transformation
: x ↦−→ xk + εk(x − xk)
satisfies (B1(xk)) = Bεk(xk) and (B1(y)) ⊂ Bεk(xk + εk(y − xk)) so that
|∇(φτk ◦ ατk)(x)|
Bε (xk) k p dx = ε 2
|∇(φτk k
◦ ατk)
B1(xk)
xk + εk(z − xk) | p dz
= ε 2
k ε −p
k |∇ξk(z)| p dz
and
B1(xk)
∇ξkL p (B1(xk)) = ε 1−(2/p)
k ∇(φτk ◦ ατk)L p (Bε k (xk)) (3.15)
= 1
by (3.14) and, for any y for which ξk|B1(y) is defined and for large enough k, wehave
∇ξkL p (B1(y)) ≤ ε 1−(2/p)
k ∇(φτk ◦ ατk)L p (Bε k (xk+εk(y−xk))) ≤ 1 (3.16)
by (3.13). The functions ξk satisfy the equation
¯∂ξk(z) = εk ˆFτk
xk + εk(z − xk) + εk ˆGτk
xk + εk(z − xk) =: Hτk(z) (3.17)

72 CASIM ABBAS
and, for every R>0, wehave
sup ∇ξkL
k
p (BR(0)) < ∞, ∇ξkL p (B2(0)) ≥ 1 (3.18)
because of (3.16) and(3.15) sinceB1(xk) ⊂ B2(0) for large k. The upper bound
on ∇ξkL p (BR(0)) depends on how many balls B1(y) are needed to cover BR(0). We
compute for ρ>0
HτkLp (Bρ(xk)) = ε 1−(2/p)
k ˆHτkL p (Bρε (xk)),
k
with ˆHτk as in (3.8). We conclude from p>2, (3.9), and (3.10) that
for any R>0 as k →∞. Defining
HτkL p (BR(0)) −→ 0
X l,p :={ψ ∈ W l,p (B,C 2 ) | ψ(0) = 0, ψ(∂B) ⊂ R 2 },l≥ 1, B⊂ C aball,
the Cauchy-Riemann operator
¯∂ : X l,p −→ W l−1,p (B,C 2 )
is a bounded bijective linear map. By the open mapping principle, we have the following
estimate:
ψl,p,B ≤ C ∂ψl−1,p,B, ∀ ψ ∈ X l,p . (3.19)
Let R ′ ∈ (0,R). Now pick a smooth function β : R 2 → [0, 1] with β|B R ′ (0) ≡ 1 and
supp(β) ⊂ BR(0). Define
We note that
ζk(z) := Re ξk(z) − ξk(0) + iβ(z) Im ξk(z) − ξk(0) .
sup Im(ξk)L
k
p (BR(0)) ≤ CR
with a suitable constant CR > 0 because of the uniform bound
sup Im(ξk)L
k,R
∞ (BR(0)) < ∞.

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 73
Using (3.19), we then obtain
because of
ξk − ξk(0)1,p,B R ′ (0) ≤ζk1,p,BR(0)
≤ C ∂ζkL p (BR(0))
≤ CR ′(HτkL p (BR(0)) +∇ξkL p (BR(0))
+Im(ξk) − Im(ξk)(0)L p (BR(0)))
∂ζk = Hτk + i(β − 1)∂ Im(ξk) + i ∂β Im(ξk) − Im(ξk)(0) .
(3.20)
Hence the sequence (ξk − ξk(0)) is uniformly bounded in W 1,p (BR ′(0)), and in particular,
it has a subsequence which converges in Cα 2
(BR ′(0)) for 0

74 CASIM ABBAS
W 1,p
loc (C) since the sequence (ξk − ξk) depends on the choice of the ball BR(0). Our
sequence ξk − ξk(0) has a convergent subsequence on any ball.
3.4. Convergence in W 1,p (B ′ )
Pick a sequence τk ↗ τ0. We claim that the sequence ( ˆFτk) converges in L p (B) maybe
after passing to a suitable subsequence (recall that, so far, we only have the uniform
bound (3.9)). The functions ˆFτk converge pointwise almost everywhere after passing
to some subsequence. Indeed, the sequence {(u∗ 0λ ◦ jτk − i(u∗ 0λ))ατ (z)} converges
k
already pointwise since jτk and ατk do (recall that the sequence (ατk) converges in
W 1,p (B) and therefore uniformly). The sequence (∂sατk) converges in Lp (B) and
therefore pointwise almost everywhere after passing to a suitable subsequence. Then
by Egorov’s theorem, for any δ>0, there is a subset Eδ ⊂ B with |B\Eδ| ≤δ so that
the sequence ˆFτk converges uniformly on Eδ. Letαbe the Lp-limit of the sequence
(∂sατk), andletε>0. We introduce
C := 2 sup (u
0≤τ0 sufficiently small such that
αL p (B\Eδ) ≤ ε
3 C .
Now choose k0 ≥ 0 so large that, for all k ≥ k0, wehave
∂sατk − αL p (B) ≤ ε
3 C
Then, if k, l ≥ k0, wehave
and ˆFτk − ˆFτlL ∞ (Eδ) ≤ ε
3 |B| .
ˆFτk − ˆFτlL p (B) ≤ˆFτk − ˆFτlL p (Eδ) +ˆFτk − ˆFτlL p (B\Eδ)
proving the claim.
≤|Eδ|ˆFτk − ˆFτlL ∞ (Eδ) + 2 sup ˆFτkL
k≥k0
p (B\Eδ)
≤|B|ˆFτk − ˆFτl p
L ∞ (Eδ)
≤ ε
+ C · sup ∂sατkL
k≥k0
p (B\Eδ)
Recalling that φτ = aτ + ifτ and that the family fτ satisfies a uniform L ∞ -bound, we
have
sup Im(φτ ◦ ατ )L
τ
∞ (B) < ∞.

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 75
Now pick three balls B ′′′ ⊂ B ′′ ⊂ B ′ ⊂ B such that the closure of one is contained
in the next. Our aim is to establish W 1,p (B ′′′ )-convergence of a subsequence of the
sequence (φτk ◦ ατk). By Proposition 3.10, we have a uniform Lp (B ′ )-bound on the
gradient. If β : R2 → [0, 1] is a smooth function with supp (β) ⊂ B ′ and β|B ′′ ≡ 1
and if
ζτ = Re φτ ◦ ατ − φτ (0) + iβ Im φτ ◦ ατ − φτ (0) ,
then we proceed in the same way as in (3.20), and we obtain
ϕk1,p,B ′′ ≤ C ( ˆHτkL p (B ′ ) +∇(φτk ◦ ατk)L p (B ′ ) +Im(ϕk)L p (B ′ )),
where we wrote
ϕk := φτk ◦ ατk − (φτk ◦ ατk)(0),
and where C>0 is a constant depending only on p, B ′ ,andB ′′ . The sequence (ϕk) is
then uniformly bounded in W 1,p (B ′′ ) by Proposition 3.10, and it converges in L p (B ′′ )
after passing to a suitable subsequence. We now use the regularity estimate
ϕl − ϕk1,p,B ′′′ ≤ C ( ˆFτl − ˆFτkL p (B ′′ ) (3.22)
+ˆGτl − ˆGτkL p (B ′′ ) +ϕl − ϕkL p (B ′′ )).
Since the right-hand side converges to zero as k, l →∞, we obtain the following.
PROPOSITION 3.12
For every ball B ′ ⊂ B ′ ⊂ B, the sequence (φτk ◦ ατk − φτk(0)) has a subsequence
which converges in W 1,p (B ′ ).
3.5. Improving regularity using both the Beltrami and Cauchy-Riemann equations
In order to improve the convergence of the conformal transformations ατk , we need
to improve the convergence of the maps µτk → µτ0 and the regularity of its limit.
It is known that the inverses α−1 of the conformal transformations ατk also satisfy a
τk
Beltrami equation (see [9])
where
∂α −1
τ = ντ ∂α −1
τ ,
ντ (z) =− ∂ατ (α −1
τ (z))
∂ατ (α −1
τ (z))µτ
α −1
τ (z)

76 CASIM ABBAS
(follows from 0 = ∂(α−1 τ ◦ ατ ) = ∂α−1 τ (ατ )∂ατ + ∂α−1 τ (ατ )∂ατ ). After passing to a
suitable subsequence, we may assume that ∂ατk and ∂ατk converge pointwise almost
everywhere since they converge in Lp (B). Hence we may assume that the sequence
(ντk) also converges pointwise almost everywhere. We also have
|ντ (z)| ≤
µτ
−1
α (z) ,
and hence ντ satisfies the same L ∞ -bound as µτ . By Lemma 3.6, we conclude that
α −1
τk
−→ α−1
1 in W 1,p (B)
with the same p>2 as in Lemma 3.6 applied to the functions µτ . After passing to
some subsequence, the sequence
(ϕk) := φτk − aτk(0) ◦ ατk
converges in W 1,p (B ′ ) for any ball B ′ ⊂ B by Proposition 3.12. Indeed, the expression
φτk ◦ ατk − φτk(0) and ϕk differ by a constant term ifτk(0), but the sequence (ifτk(0))
has a convergent subsequence.
We would like to derive a decent notion of convergence for the sequence (ϕτk◦α −1
τk ),
but the space W 1,p is not well behaved under compositions. The composition of two
functions of class W 1,p is only in W 1,p/2 . Since we cannot choose p>2 freely, we
rather carry out the argument in Hölder spaces. By the Sobolev embedding theorem
and Rellich compactness, we may assume that the sequences (ϕk) and (α−1) converge
τk
in C0,α (B ′ ) for any ball B ′ ⊂ B ′ ⊂ B and 0

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 77
spaces, that is,
ϕl − ϕkC k+1,γ (B ′′′ ) ≤ C ( ˆFτl − ˆFτkC k,γ (B ′′ )
+ˆGτl − ˆGτkC k,γ (B ′′ ) +ϕl − ϕkC k,γ (B ′′ )).
The sequence ( ˆFτk) now converges in the C0,α2-norm, and the sequence ( ˆGτk) converges
in any Hölder norm. We obtain with the above regularity estimate C1,α2-convergence of
the sequence (ϕk), and composing with α−1 yields C1,α4-convergence
of (fτk) and (µτl).
τk
Invoking Theorem 3.7 again then improves the convergence of the transformations
ατk, α−1 to C τk 2,α4.
We now iterate the procedure using the regularity estimate for the
Cauchy-Riemann operator in Hölder space and the estimate for the Beltrami equation
in Theorem 3.7.
Theorem 3.2 follows if we apply the implicit function theorem to the limit solution
(S,jτ0, ũτ0 = (aτ0,uτ0),γτ0), hence we obtain the same limit for every sequence {τk},
and we obtain convergence in C∞ .
4. Conclusion
The following remarks tie together the loose ends and prove the main result, Theorem
1.6. We start with a closed 3-dimensional manifold with contact form λ ′ . Giroux’s
theorem, Theorem 1.4, then permits us to change the contact form λ ′ to another
contact form λ such that ker λ = ker λ ′ and such that there is a supporting open book
decomposition with binding K consisting of periodic orbits of the Reeb vector field
of λ. Invoking Proposition 2.4, we construct a family of 1-forms (λδ)0≤δ

78 CASIM ABBAS
is finite, the images of uτ and u0 must agree for some sufficiently large τ, concluding
the proof.
Appendix. Some local computations near the punctures
In this appendix, we present some local computations needed for the proof of Theorem
2.8. The issue is to show that the 1-forms
u ∗
0 λ ◦ jf − da0 and u ∗
0 λ + da0 ◦ jf
are bounded on ˙S. We obtain in the second case of the theorem
u ∗
0 λ ◦ jf − da0 = u ∗
0 λ ◦ (jf − jg) + γ0
= dg ◦ (jf − jg) + γ0 + v ∗
0 λ ◦ (jf − jg).
The first case can be treated as a special case: here the objective is to show that the
1-form v ∗ 0 λ ◦ (jf − i) = v ∗ 0 λ ◦ (jf − j0) is bounded near the punctures. We again drop
the subscript δ in the notation since we are only concerned with a local analysis near
the binding, and all the forms λδ are identical there. We use coordinates (θ,r,φ) near
the binding. The contact structure is then generated by
η1 = ∂
∂r = (0, 1, 0), η2
∂
=−γ2
∂θ
+ γ1
∂
∂φ = (−γ2, 0,γ1).
The projection onto the contact planes along the Reeb vector field is then given by
πλ(v1,v2,v3) = 1
µ (v1γ ′ ′
1 + v3γ 2 ) η2 + v2 η1, with µ = γ1γ ′ ′
2 − γ 1γ2, and the flow of the Reeb vector field is given by
where
φt(θ,r,φ) = θ + α(r)t,r,φ + β(r)t ,
α(r) = γ ′ 2 (r)
µ(r)
and β(r) =− γ ′ 1 (r)
µ(r) .
The linearization of the flow Tφτ (θ,r,φ) preserves the contact structure. In the basis
{η1,η2} it is given by
1 0
Tφτ (θ,r,φ) =
with A(r) =
τA(r) 1
1
µ 2 ′′ ′ ′′ ′
γ 2 (r)γ 1 (r) − γ 1 (r)γ 2
(r)
(r) .

HOLOMORPHIC OPEN BOOK DECOMPOSITIONS 79
The complex structure(s) we chose earlier in (2.7) had the following form near the
binding with respect to the basis {η1,η2}:
J (θ,r,φ) =
0 −rγ1(r)
0
.
1
rγ1(r)
The induced complex structure jτ on the surface is then given by
jτ (z) = [πλTv0(z)] −1
[Tφτ v0(z) ] −1 J φτ (v0(z))
Tφτ v0(z) πλTv0(z).
With v0(s, t) = (t,r(s),α),wefindthat
so that
and
jτ =
πλTv0(s, t) =
r ′ (s) 0
0
γ ′ 1 (s)
µ(r(s))
−τA(r)rγ1(r) − rγ1(r)γ ′ 1 (r)
r ′ µ(r)
r ′ µ(r)
rγ1(r)γ ′ 1 (r)(1 + τ 2 A 2 (r)r 2 γ 2 1
(r)) τA(r)rγ1(r)
−τA(r)rγ1(r) −1
=
1 + τ 2A2 (r)r2γ 2
1 (r) τA(r)rγ1(r)
−1 0
= j0 + τA(r) γ1(r)
τA(r)γ1(r) 1
jτ − jσ = A(r)rγ1(r)(τ − σ )
using the fact that r(s) satisfies the differential equation
With v ∗ 0 λ = γ1(r) dt, we obtain
r ′ (s) = γ ′ 1 (r(s))γ1(r(s))r(s)
.
µ(r(s))
−1 0
(τ + σ )A(r)rγ1(r) 1
v ∗
0λ ◦ (jτ − jσ )|(s,t) = (τ − σ )A r(s) r(s)γ 2
1 r(s)
· (τ + σ )A r(s)
r(s)γ1 r(s) ds + dt .
,

80 CASIM ABBAS
Converting from coordinates (s, t) on the half-cylinder to Cartesian coordinates x +
iy = e −(s+it) in the complex plane, we get, with ρ = x 2 + y 2 ,
ds =− 1
(xdx+ ydy) and dt =−1 (xdy− ydx).
ρ2 ρ2 Recall that r(s) = c(s)e κs , where c(s) is a smooth function which converges to a
constant as s →+∞, and we assumed that κ ≤−1/2 and that κ /∈ Z. Another
assumption was that A(r) = O(r). Hence r(s) is close to ρ −κ if s is large (and ρ is
small) and A(r(s)) = O(ρ −κ ). Also recall that γ1(r(s)) = O(1). Summarizing, we
need the expression
A r(s) r(s)ρ −2 ρ = O(ρ −2κ−1 )
to be bounded, which amounts to κ ≤−1/2. The same argument applied to the form
dg ◦ (jτ − jσ ) leads to the same conclusion.
Acknowledgments. I am very grateful to Richard Siefring and Chris Wendl for explaining
some of their work to me. Their results are indispensable for the arguments
in this article. I would also like to thank Samuel Lisi for the numerous discussions we
had about the subject of this article.
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Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA;
abbas@math.msu.edu

CALDERÓN INVERSE PROBLEM WITH PARTIAL
DATA ON RIEMANN SURFACES
COLIN GUILLARMOU and LEO TZOU
Abstract
On a fixed smooth compact Riemann surface with boundary (M0,g), we show that,
for the Schrödinger operator g + V with potential V ∈ C 1,α (M0) for some α>0,
the Dirichlet-to-Neumann map N |Ɣ measured on an open set Ɣ ⊂ ∂M0 determines
uniquely the potential V . We also discuss briefly the corresponding consequences for
potential scattering at zero frequency on Riemann surfaces with either asymptotically
Euclidean or asymptotically hyperbolic ends.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2. Harmonic and holomorphic Morse functions on a Riemann surface . . . 86
3. Carleman estimate for harmonic weights with critical points . . . . . . . 95
4. Complex geometric optics on a Riemann surface . . . . . . . . . . . . . 99
5. Identifying the potential . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6. Inverse scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Appendix. Boundary determination . . . . . . . . . . . . . . . . . . . . . . 114
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
1. Introduction
The problem of determining the potential in the Schrödinger operator by boundary
measurement goes back to Calderón [8]andGelfand[12]. Mathematically, it amounts
to asking if one can detect some data from boundary measurement in a domain (or
manifold) with boundary. The typical model to have in mind is the Schrödinger
operator P := g + V , where g is a metric and V is a potential; then we define the
Cauchy data space by
C := (u|∂,∂νu|∂); u ∈ H 1 (), u∈ ker P ,
DUKE MATHEMATICAL JOURNAL
Vol. 158, No. 1, c○ 2011 DOI 10.1215/00127094-1276310
Received 17 September 2009. Revision received 22 July 2010.
2010 Mathematics Subject Classification. Primary 35R30; Secondary 58J32.
Guillarmou’s work partially supported by National Science Foundation grant DMS-0635607.
Tzou’s work partially supported by National Science Foundation grant DMS-0807502.
83

84 GUILLARMOU and TZOU
where ∂ν is the interior pointing normal vector field to ∂ and where H 1 () =
W 1,2 () is the space of L2 functions with one derivative in L2 .
The first natural question is the following full data inverse problem: does the
Cauchy data space determine uniquely the metric g and/or the potential V ?Ina
sense, the most satisfying known results are those obtained when the domain ⊂ Rn is already known and g is the Euclidean metric, for which the identification of V
has been proved in dimension n>2 by Sylvester and Uhlmann [32] and Novikov
[30] (where reconstruction is also obtained), and very recently in dimension 2 by
Bukgheim [6] when the domain is simply connected. In a recent work [16], we
proved the identification of V on a Riemann surface with boundary from full data
measurement. A related question is the conductivity problem which consists in taking
V = 0 and replacing g by −divσ ∇, where σ is a positive definite symmetric tensor.
An elementary observation shows that the problem of recovering a sufficiently smooth
isotropic conductivity (i.e., σ = σ0Id for a function σ0) is contained in the abovementioned
problem of recovering a potential V . For domain of R2 , Nachman [29]
used the ¯∂ techniques to show that the Cauchy data space determines the conductivity.
Recently a new approach developed by Astala and Päivärinta in [2] improved this
result to assuming that the conductivity is only an L∞ scalar function on a simply
connected domain. This was later generalized to L∞ anisotropic conductivities by
Astala, Lassas, and Päivärinta in [3]. We notice that there still are rather few results
in the direction of recovering the Riemannian manifold (,g) when V = 0, for
instance the surface case by Lassas and Uhlmann [24] (seealso[4], [17]), the real
analytic manifold case by Lassas, Taylor, and Uhlmann [23] (seealso[15] forthe
Einstein case), the case of manifolds admitting limiting Carleman weights and in a
same conformal class by Dos Santos Ferreira, Kenig, Salo, and Uhlmann [9]. The
second natural, but harder, problem is the partial data inverse problem: if Ɣ1 and Ɣ2
are open subsets of ∂, does the partial Cauchy data space for P
CƔ1,Ɣ2 := (u|Ɣ1,∂νu|Ɣ2); u ∈ H 1
(∂M0), Pu= 0, u= 0in∂ \ Ɣ1
determine the domain , the metric, the potential? For a fixed domain of R n ,the
recovery of the potential if n>2 with partial data measurements was initiated by
Bukhgeim and Uhlmann [7] and later improved by Kenig, Sjöstrand, and Uhlmann
[21] to the case where Ɣ1 and Ɣ2 are respectively open subsets of the “front” and
“back” ends of the domain. We refer the reader to the references for a more precise
formulation of the problem. In dimension 2, the recent works of Imanuvilov, Uhlmann,
and Yamamoto [19] solves the problem for fixed domains of R 2 in the case when
Ɣ1 = Ɣ2 and when the potential are in C 2,α () for some α>0. In this article, we
address the same question when the background domain is a fixed Riemann surface
with boundary. We prove the following recovery result.

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 85
THEOREM 1.1
Let (M0,g) be a smooth compact Riemann surface with boundary, let Ɣ ⊂ ∂M0 be
an open subset of the boundary, and let g be the nonnegative Laplacian on M0.For
α ∈ (0, 1),letV1,V2 ∈ C1,α (M0) be two potentials and, for i = 1, 2,let
C Ɣ
i =: (u|Ɣ,∂νu|Ɣ); u ∈ H 1 (M0), (g + Vi)u = 0, u = 0on∂M0 \ Ɣ
(1)
be the respective Cauchy partial data spaces. If C Ɣ 1 = C Ɣ 2 ,thenV1 = V2.
Here the space C1,α (M0) is the usual Hölder space for α ∈ (0, 1). Notice that when
g + Vi do not have L2 eigenvalues for the Dirichlet condition, the statement above
can be given in terms of Dirichlet-to-Neumann operators. Since ˆg = e−2ϕg when
ˆg = e2ϕg for some function ϕ, it is clear that in the statement in Theorem 1.1, we
need only to fix the conformal class of g instead of the metric g (or equivalently,
we need only to fix the complex structure on M0). In particular, the smoothness
assumption of the Riemann surface with boundary is not really essential since we
can change it conformally to make it smooth; for the Cauchy data space, this just
has the effect of changing the potential conformally (all we need is that this new
potential be C1,α ). Observe also that Theorem 1.1 implies that, for a fixed Riemann
surface with boundary (M0,g), the Dirichlet-to-Neumann map on Ɣ for the operator
u →−divg(γ ∇gu) determines the isotropic conductivity γ if γ ∈ C3,α (M0) in the
sense that two conductivities giving rise to the same Dirichlet-to-Neumann map are
equal. This is a standard observation by transforming the conductivity problem to a
potential problem with potential V := (gγ 1/2 )/γ 1/2 . So our result also extends that
of Henkin and Michel [17] in the case of isotropic conductivities. Notice also that
reconstruction methods for isotropic conductivities are obtained in a recent work of
Henkin and Novikov [18] for full boundary data measurements on Riemann surfaces.
The method to identify the potential follows [6] and[19] and is based on the
construction of a large set of special complex geometric optic solutions of (g +
V )u = 0, also called Faddeev-type solutions and introduced first by Faddeev in [10].
More precisely, if Ɣ0 = ∂M0 \ Ɣ is the set where we do not know the Dirichletto-Neumann
operator, then we construct solutions of the form u = Re e/h (a +
r(h)) + eRe()/hs(h) with u|Ɣ0 = 0, where h>0 is a small parameter, and a
are holomorphic functions on (M0,g) independent of h,and||r(h)||L2 = O(h) while
||s(h)||L2 = O(h3/2 | log h|) as h → 0. The idea of [6] to reconstruct V (p) for p ∈ M0
is to take with a nondegenerate critical point at p and then use stationary phase as
h → 0. In our setting, the function needs to be purely real on Ɣ0 and Morse with a
prescribed critical point at p. One of our main contributions is a geometric construction
of the holomorphic Carleman weights satisfying such conditions. We should point
out that we use a method quite different from that in [19] to construct this weight,
and we believe that our method simplifies their construction even in their case. A

86 GUILLARMOU and TZOU
Carleman estimate on the surface for this degenerate weight needs to be proved, and
we follow the ideas of [19]. We manage to improve the regularity of the potential to
C1,α instead of C2,α in [19]. Since it did not seem to be available in the literature in
our geometric setting, we also provide a short proof in the appendix of the fact that the
partial Cauchy data space C Ɣ determines a potential V ∈ C0,α (M0) on Ɣ (for α>0).
An interesting fact about this proof is that it uses our Carleman estimate, and it allows
rather weak assumptions on the regularity of V .
In Section 6, we obtain two inverse scattering results as a corollary of Theorem
1.1, first for partial data scattering at zero frequency for + V on asymptotically
hyperbolic surfaces with potential decaying at the conformal infinity, and second for
full data scattering at zero frequency for + V with V compactly supported on an
asymptotically Euclidean surface.
Another straightforward corollary in the asymptotically Euclidean case full data
setting is the recovery of a compactly supported potential from the scattering operator
at a positive frequency. The proof is essentially the same as for the operator Rn + V
once we know Theorem 1.1, so we omit it.
2. Harmonic and holomorphic Morse functions on a Riemann surface
2.1. Riemann surfaces
We start by recalling a few elementary definitions and results about Riemann surfaces
(see, e.g., [11] for more details). Let (M0,g0) be a compact, oriented, connected,
smooth Riemannian surface with boundary ∂M0. The surface M0 can be considered
as a subset of a larger oriented Riemannian surface with boundary (M,g), extending
smoothly the metric g0 to M. The conformal class of g and the orientation on
the surface M induce a structure of Riemann surface with boundary (i.e., a surface
equipped with a complex structure via holomorphic charts zα : Uα → C). The Hodge
star operator ⋆ (well defined since M is oriented) acts on the cotangent bundle T ∗M, its eigenvalues are ±i, and the respective eigenspaces T ∗
1,0
M := ker(⋆ + iId) and
T ∗
0,1 M := ker(⋆ − iId) are subbundles of the complexified cotangent bundle CT ∗ M,
and the splitting CT ∗M = T ∗
∗
1,0M ⊕ T0,1M holds as complex vector spaces. Since ⋆
is conformally invariant on 1-forms on M, the complex structure depends only on the
conformal class of g. In holomorphic coordinates z = x + iy in a chart Uα, one has
⋆(udx + vdy) =−vdx + udy and
T ∗
1,0 M|Uα C dz, T ∗
0,1 M|Uα C d ¯z,
where dz = dx + idy and d ¯z = dx − idy. We define the natural projections induced
by the splitting of CT ∗ M as
π1,0 : CT ∗ M → T ∗
1,0 M, π0,1 : CT ∗ M → T ∗
0,1 M.

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 87
The exterior derivative d defines the de Rham complex 0 → 0 → 1 → 2 → 0,
where k := kT ∗M denotes the real bundle of k-forms on M. Let us denote Ck the complexification of k . Then the ∂ and ¯∂ operators can be defined as differential
operators ∂ : C 0 → T ∗
1,0 M and ¯∂ : C0 → T ∗
0,1
M by
∂f := π1,0 df, ¯∂ := π0,1 df, (2)
where they satisfy d = ∂ + ¯∂ and where they are expressed in holomorphic coordinates
by
∂f = ∂zf dz,
¯∂f = ∂¯zf d¯z,
with ∂z := (∂x − i∂y)/2 and ∂¯z := (∂x + i∂y)/2. Similarly, one can define the ∂ and
¯∂ operators from C 1 to C 2 by setting
∂(ω1,0 + ω0,1) := dω0,1,
¯∂(ω1,0 + ω0,1) := dω1,0
if ω0,1 ∈ T ∗
0,1M and ω1,0 ∈ T ∗
1,0M. In coordinates, this is simply
∂(udz+ vd¯z) = ∂v ∧ d ¯z,
¯∂(udz+ vd¯z) = ¯∂u ∧ dz.
There is a natural operator, the Laplacian acting on functions and defined by
f := −2i ⋆¯∂∂f = d ∗ d,
where d ∗ is the adjoint of d through the metric g and where ⋆ is the Hodge star operator
mapping 2 to 0 and induced by g as well.
2.2. Maslov index and boundary-value problem for the ∂ operator
In this section, we consider the setting where M0 is an oriented Riemann surface with
boundary ∂M0 and Ɣ ⊂ ∂M0 is an open subset, and we let Ɣ0 = ∂M0 \ Ɣ be its
complement in ∂M0. We can identify the boundary with a disjoint union of circles
∂M0 = m i=1 ∂iM0, where each ∂iM0 S1 .SinceƔwill be the piece of the boundary
where we know the Cauchy data space, it is sufficient to assume that Ɣ is a connected
nonempty open segment of ∂1M0 = S1 , which we now do. Following [26], we adopt
the following notation: let E → M0 be a complex line bundle with complex structure
J : E → E, andletD : C∞ (M0,E) → C∞ (M0,T∗ 0,1 ⊗ E) be a Cauchy-Riemann
operator with smooth coefficients on M0 acting on sections of the bundle E. Observe
that in the case when E = M0 × C is the trivial line bundle with the natural complex
structure on M0,thenDcan be taken to be the operator ∂ introduced in (2). For q>1,
we define
DF : W ℓ,q
F (M0,E) → W ℓ−1,q (M0,T ∗
0,1 M0 ⊗ E),

88 GUILLARMOU and TZOU
where F ⊂ E |∂M0 is a totally real subbundle (i.e., a subbundle such that JF ∩ F is
the zero section) and where DF is the restriction of D to the Lq-based Sobolev space
with ℓ derivatives and boundary condition F
W ℓ,q
F (M0,E):= ξ ∈ W ℓ,q (M0,E) ξ(∂M0) ⊂ F .
When q = 2, we will use the standard notation H ℓ (M0) := W ℓ,2 (M0) for L 2 -based
Sobolev spaces. The boundary Maslov index for a totally real subbundle F ⊂ E∂M0
of a complex vector bundle is defined in generality in [26, Appendix C.3]; we only
recall the definition for our setting.
Definition 2.2.1
Let E = M0 × C and ∂M0 = m j=1 ∂iM0 be a disjoint union of m circles. The
boundary Maslov index µ(E,F) is the degree of the map ρ ◦ : ∂M0 → ∂M0,
where
|∂iM0 : S 1 ∂iM0 → GL(1, C)/GL(1, R)
is the natural map assigning to z ∈ S 1 the totally real subspace Fz ⊂ C, where
GL(1, C)/GL(1, R) is the space of totally real subbundles of C, and where ρ :
GL(1, C)/GL(1, R) → S 1 is defined by ρ(A.GL(1, R)) := A 2 /|A| 2 .
In this setting, we have the following boundary-value Riemann-Roch theorem stated
in [26, Theorem C.1.10].
THEOREM 2.2.2
Let E → M0 be a complex line bundle over an oriented compact Riemann surface
with boundary, and let F ⊂ E |∂M0 be a totally real subbundle. Let D be a smooth
Cauchy-Riemann operator on E acting on W ℓ,q (M0,E) for some q ∈ (1, ∞) and
ℓ ∈ N. Then we have the following.
(1) The following operators are Fredholm:
DF : W ℓ,q
F (M0,E) → W ℓ−1,q (M0,T ∗
0,1 M0 ⊗ E),
D ∗
F
ℓ,q ∗
: WF (M0,T0,1M0 ⊗ E) → W ℓ−1,q (M0,E).
(2) The real Fredholm index of DF is given by
Ind(DF ) = χ(M0) + µ(E,F),
where χ(M0) is the Euler characteristic of M0 and where µ(E,F) is the
boundary Maslov index of the subbundle F .

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 89
(3) If µ(E,F) < 0, thenDF is injective, while if µ(E,F) + 2χ(M0) > 0, the
operator DF is surjective.
As an application, we obtain the following (here and in what follows, H m (M0) :=
W m,2 (M0)).
COROLLARY 2.2.3
(i) For q > 1 and k ∈ N0, letω∈ W k,q (M0,T∗ 0,1M0). Then there exists u ∈
W k+1,q (M0) real-valued on Ɣ0 such that ¯∂u = ω.
(ii) For m>1/2, letf∈ H m (∂M0) be a real-valued function. Then there exists
a holomorphic function v ∈ H m+(1/2) (M0) such that Re(v)|Ɣ0 = f .
(iii) For k ∈ N and q>1, the space of W k,q (M0) holomorphic functions on M0
which are real-valued on Ɣ0 is infinite-dimensional.
Proof
(i) Let L ∈ N be arbitrarily large, and let Ɣ be a connected nonempty open segment
of one connected component ∂1M0 = S 1 of ∂M0. ThenƔ can be defined in a
coordinate θ (respecting the orientation of the boundary) by Ɣ ={θ ∈ S 1 |
0 0.SinceL can be taken as
large as we want, this achieves the proof of (i).
(ii) Let w ∈ H m+(1/2) (M0) be a real function with boundary value f on ∂M0.Then
by (i), there exists R ∈ H m+(1/2) (M0) such that i ¯∂R =−¯∂w and R purely real
on Ɣ0; thus v := iR + w is holomorphic such that Re(v) = f on Ɣ0.
(iii) Taking the subbundle F as in the proof of (i), we have that dim ker DF =
χ(M0) + 2L if L satisfies 2χ(M0) + 2L >0, and since L can be taken as large
as we like, this concludes the proof.

90 GUILLARMOU and TZOU
LEMMA 2.2.4
Let {p0,p1,...,pn} ⊂M0 be a set of n + 1 disjoint points. Let c1,...,cK ∈ C, let
N ∈ N, and let z be a complex coordinate near p0 such that p0 ={z = 0}. Thenif
p0 ∈ int(M0), there exists a holomorphic function f on M0 with zeros of order at least
N at each pj such that f is real on Ɣ0 and f (z) = c0 +c1z+···+cKz K +O(|z| K+1 )
in the coordinate z.Ifp0 ∈ ∂M0, then the same is true except that f is not necessarily
real on Ɣ0.
Proof
First, using linear combinations and induction on K, it suffices to prove the lemma for
any K and c0 =···=cK−1 = 0, which we now show. Consider the subbundle F as
in the proof of (i) in Corollary 2.2.3. The Maslov index µ(E,F) is given by 2L, and so
for each N ∈ N, one can take L large enough to have µ(F,E)+2χ(M0) ≥ 2N(1+n).
Therefore, by Theorem 2.2.2, the dimension of the kernel of ∂F will be greater than
2(n + 1)N. Now, since for each pj and complex coordinate zj near pj the map
u → (u(pj),∂zju(pj),...,∂N−1 zj u(pj)) ∈ CN is linear, this implies that there exists
a nonzero element u ∈ ker DF which has zeros of order at least N at all pj.
First, assume that p0 ∈ int(M0) and that we want the desired Taylor expansion
at p0 in the coordinate z. In the coordinate z, one has u(z) = αzM + O(|z| M+1 ) for
some α = 0 and M ≥ N. Define the function rK(z) = χ(z) cK
α z−M+K , where χ(z) is
a smooth cutoff function supported near p0 and which is 1 near p0 ={z = 0}. Since
M ≥ N>1, this function has a pole at p0 and trivially extends smoothly to M0\{p0},
which we still call rK. Observe that the function is holomorphic in a neighborhood
of p0 but not at p0 where it is only meromorphic, so that in M0 \{p0}, ∂rK is a
smooth and compactly supported section of T ∗
0,1 M0, and therefore trivially extends
smoothly to M0 (by setting its value to be zero p0) to a 1-form denoted ωK. Bythe
surjectivity assertion in Corollary 2.2.3, there exists a smooth function RK satisfying
∂RK =−ωK,andRK|Ɣ0 ∈ R.WenowhavethatRK + rK is a holomorphic function
on M\{p0} meromorphic with a pole of order M − K at p0, and in coordinate z one
has z M−K (RK(z) + rK(z)) = cK + O(|z|). Setting f = u(RK + rK), wehavethe
desired holomorphic function. Note that f also vanishes to order N at all p1,...,pn
since u vanishes. This achieves the proof.
Now, if p0 ∈ ∂M0, we can consider a slightly larger smooth domain of M containing
M0, and we apply the result above.
2.3. Morse holomorphic functions with prescribed critical points
The main result of this section is the following.
PROPOSITION 2.3.1
Let p be an interior point of M0, and let ɛ > 0 be small. Then there exists a
holomorphic function on M0 which is Morse on M0 (up to the boundary) and

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 91
real-valued on Ɣ0, which has a critical point p ′ at distance less than ɛ from p and
such that Im((p ′ )) = 0.
Let O be a connected open set of M such that Ō is a smooth surface with boundary,
with M0 ⊂ Ō ⊂ M and Ɣ0 ⊂ ∂Ō.Fixk>2 a large integer; we denote by Ck ( Ō) the
Banach space of Ck real-valued functions on Ō. Then the set of harmonic functions
on Ō which are in the Banach space Ck ( Ō) (and smooth in O by elliptic regularity)
is the kernel of the continuous map : Ck ( Ō) → Ck−2 ( Ō), and so it is a Banach
subspace of Ck ( Ō). The set H ⊂ Ck ( Ō) of harmonic functions u in Ck ( Ō) such that
there exists v ∈ Ck ( Ō) harmonic with u + iv holomorphic on O is a Banach subspace
of Ck ( Ō) of finite codimension. Indeed, let {γ1,...,γN} be a homology basis for O;
then we have
defined by
H = ker L with L :ker∩C k ( Ō) → CN
L(u) :=
1
∂u .
πi γj
j=1,...,N
For all Ɣ ′ 0 ⊂ ∂M0 such that the complement of Ɣ ′ 0
We now show the following.
LEMMA 2.3.2
HƔ ′ 0 :={u ∈ H; u|Ɣ ′ 0
contains an open subset, we define
= 0}.
The set of functions u ∈ HƔ ′ 0
intersection of open dense sets) in HƔ ′ 0 with respect to the Ck (
which are Morse in O is residual (i.e., a countable
Ō) topology.
Proof
We use an argument very similar to that used by Uhlenbeck [34]. We start by defining
m : O × HƔ ′ 0 → T ∗O by (p, u) ↦→ (p, du(p)) ∈ T ∗ p O. This is clearly a smooth map,
linear in the second variable, and moreover mu := m(., u) = (·,du(·)) is Fredholm
since O is finite-dimensional. The map u is a Morse function if and only if mu is
transverse to the zero section, denoted T ∗
0 O,ofT∗O, that is, if
Image(Dpmu) + Tmu(p)(T ∗
0 O) = Tmu(p)(T ∗ O) ∀p ∈ O such that mu(p) = (p, 0),
which is equivalent to the fact that the Hessian of u at critical points is nondegenerate
(see, e.g., [34, Lemma 2.8]). We recall the following transversality theorem.

92 GUILLARMOU and TZOU
THEOREM 2.3.3 ([34, Theorem 2])
Let m : X × HƔ ′ 0 → W be a Ck map, where X, HƔ ′ , and W are separable Banach
0
manifolds with W and X of finite dimension. Let W ′ ⊂ W be a submanifold such
that k>max(1, dim X − dim W + dim W ′ ).Ifmis transverse to W ′ , then the set
{u ∈ HƔ ′ 0 ; mu is transverse to W ′ } is dense in HƔ ′ , and more precisely, it is a residual
0
set.
We want to apply it with X := O, W := T ∗O,andW ′ := T ∗
0 O, and the map m is
defined above. We have thus proved Lemma 2.3.2 if one can show that m is transverse
to W ′ .Let(p, u) be such that m(p, u) = (p, 0) ∈ W ′ . Then, identifying T(p,0)(T ∗O) with TpO ⊕ T ∗ p O, one has
D(p,u)m(z, v) = z, dv(p) + Hessp(u)z ,
where Hesspu is the Hessian of u at the point p, viewed as a linear map from
TpO to T ∗ p O. To prove that m is transverse to W ′ , we need to show that (z, v) →
(z, dv(p) + Hessp(u)z) is surjective from TpO ⊕ HƔ ′ 0 to TpO ⊕ T ∗ p O,whichis
realized, for instance, if the map v → dv(p) from HƔ ′ 0 to T ∗ p O is surjective. But from
Lemma 2.2.4, we know that there exist holomorphic functions v and ˜v on O such that
v and ˜v are purely real on Ɣ ′ 0 . Clearly, the imaginary parts of v and ˜v belong to HƔ ′ 0 .
Furthermore, for a given complex coordinate z near p ={z = 0}, we can arrange
them to have series expansion v(z) = z + O(|z| 2 ) and ˜v(z) = iz + O(|z| 2 ) around
the point p. We see, by coordinate computation of the exterior derivative of Im(v) and
Im(v ), thatdIm(v)(p) and d Im(˜v)(p) are linearly independent at the point p. This
shows our claim and ends the proof of Lemma 2.3.2 by using Theorem 2.3.3.
We now proceed to show that the set of all functions u ∈ HƔ ′ 0
degenerate critical points on Ɣ ′ 0 is also residual.
LEMMA 2.3.4
For all p ∈ Ɣ ′ 0 and k ∈ N, there exists a holomorphic function u ∈ Ck (
Im(u)|Ɣ ′ 0
= 0 and ∂u(p) = 0.
such that u has no
Ō) such that
Proof
The proof is quite similar to that of Lemma 2.2.4. By Lemma 2.2.4, we can choose a
holomorphic function v ∈ Ck ( Ō) such that v(p) = 0 and Im(v)|Ɣ ′ = 0; then either
0
∂v(p) = 0 and we are done, or ∂v(p) = 0. Assume now the second case, and let
M ∈ N be the order of p as a zero of v. By the Riemann mapping theorem, there is
a conformal mapping from a neighborhood Up of p in Ō to a neighborhood {|z| <
ɛ, Im(z) ≥ 0} of the real line Im(z) = 0 in C, and one can assume that p ={z = 0}
in these complex coordinates. Take r(z) = χ(z)z−M+1 , where χ ∈ C∞ 0 (|z| ≤ɛ) is a

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 93
real-valued function with χ(z) = 1 in {|z|

94 GUILLARMOU and TZOU
At a point (p, u) such that b(p, u) = 0, a simple computation yields that the differential
D(p,u)b : TpƔ ′ 0 × HƔ ′ 0 → T(p,∂νu(p))(N ∗ Ɣ ′ 0 ) is given by (w, u′ ) ↦→ (w, ∂τ ∂νu(p)w +
∂νu ′ (p)). This observation combined with Lemma 2.3.4 shows that, for all (p, u) ∈
Ɣ ′ 0 × HƔ ′ 0 such that b(p, u) = (p, 0), b is transverse to N ∗ 0 Ɣ′ 0 at (p, 0). Now we can
apply Theorem 2.3.3; with X = Ɣ ′ 0 , W = N ∗Ɣ ′ 0 ,andW ′ = N ∗ 0 Ɣ′ 0 , we see that the set
{u ∈ HƔ ′ 0 ; bu is transverse to N ∗ 0 Ɣ′ 0 } is residual in HƔ ′ . In view of Lemma 2.3.5, we
0
deduce the following.
LEMMA 2.3.7
The set of functions u ∈ HƔ ′ 0 such that u has no degenerate critical point on Ɣ′ 0 is
residual in HƔ ′ 0 .
Observing the general fact that finite intersection of residual sets remains residual, the
combination of Lemmas 2.3.7 and 2.3.2 yields this corollary.
COROLLARY 2.3.8
The set of functions u ∈ HƔ ′ 0
points on Ɣ ′ 0 is residual in HƔ ′ 0 with respect to the Ck (
dense.
which are Morse in O and have no degenerate critical
Ō) topology. In particular, it is
We are now in a position to give a proof of the main proposition of this section.
Proof of Proposition 2.1
As explained above, choose O in such a way that Ō is a smooth surface with boundary,
containing M0, sothatƔ0⊂∂O andsothatOcontains ∂M0\Ɣ0. LetƔ ′ 0 be an open
and such
subset of the boundary of Ō such that the closure of Ɣ0 is contained in Ɣ ′ 0
that ∂Ō\Ɣ′ 0 =∅.Letpbe an interior point of M0. By Lemma 2.2.4, there exists a
holomorphic function f = u + iv on Ō such that f is purely real on Ɣ′ 0 , v(p) = 1,
and df (p) = 0 (thus v ∈ HƔ ′ 0 ).
By Corollary 2.3.8, there exists a sequence (vj)j of Morse functions vj ∈ HƔ ′ 0
such that vj → v in C k (M0) for any fixed k large. By the Cauchy integral formula,
there exist harmonic conjugates uj of vj such that fj := uj + ivj → f in C k (M0).
Let ɛ>0 be small, and let U ⊂ O be a neighborhood containing p andnoother
critical points of f , and with boundary a smooth circle of radius ɛ. In complex local
coordinates near p, we can identify ∂f and ∂fj to holomorphic functions on an open
set of C. Then by Rouche’s theorem, it is clear that ∂fj has precisely one zero in U
and that vj never vanishes in U if j is large enough. Fix to be one of the fj for j
large enough. By construction, is Morse in O and has no degenerate critical points
on Ɣ0 ⊂ Ɣ ′ 0 . We notice that, since the imaginary part of vanishes on all of Ɣ′ 0 ,itis
clear from the reflection principle applied after using the Riemann mapping theorem

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 95
(as in the proof of Lemma 2.3.5) that no point on Ɣ0 ⊂ Ɣ ′ 0 can be an accumulation
point for critical points. Now ∂M0\Ɣ0 is contained in the interior of O, and therefore
no points on ∂M0\Ɣ0 can be an accumulation point of critical points. Since is Morse
in the interior of O, there are no degenerate critical points on ∂M0\Ɣ0. This ends the
proof.
3. Carleman estimate for harmonic weights with critical points
In this section, we prove a Carleman estimate using harmonic weight with nondegenerate
critical points in a way similar to [19]. Let = ϕ + iψ be a holomorphic Morse
function with no critical points on Ɣ0, and which is purely real on Ɣ0. In particular,
this implies that ∂νϕ|Ɣ0 = 0.
PROPOSITION 3.1
Let (M0,g) be a smooth Riemann surface with boundary, and let ϕ : M0 → R be a
C k (M0) harmonic Morse function for k large. Then for all V ∈ L ∞ (M0) there exist
C>0 and h0 > 0 such that, for all h ∈ (0,h0) and u ∈ C ∞ (M0) with u|∂M0 = 0,we
have
1
h u2
L 2 (M0)
1
+ u|dϕ|2
h2 L2 (M0) +du2 L2 2
(M0) +∂νuL2 (Ɣ0)
≤ C e −ϕ/h (g + V )e ϕ/h u 2
L2 1 2
(M0) + ∂νuL h 2
(Ɣ) ,
where ∂ν is the exterior unit normal vector field to ∂M0.
We start by modifying the weight as follows. If ϕ0 := ϕ : M0 → R is a real-valued
harmonic Morse function with critical points {p1,...,pN} in the interior of M0,then
we let ϕj : M0 → R be harmonic functions such that pj is not a critical point of
ϕj for j = 1,...,N, their existence is ensured by Lemma 2.2.4. Forallɛ>0, we
define the convexified weight ϕɛ := ϕ − (h/2ɛ)( N j=0 |ϕj| 2 ). To prove the estimate,
we will localize in charts j covering the surfaces. These charts will be taken so that
if j ∩ ∂M0 = ∅,thenj∩∂M0 S1 is a connected component of ∂M0. Moreover,
by Riemann’s mapping theorem (e.g., [25, Lemma 3.2]), this chart can be taken to be
a neighborhood of |z| =1 in {z ∈ C; |z| ≤1} and such that the metric g is conformal
to the Euclidean metric |dz| 2 .
LEMMA 3.2
Let be a chart of M0 as above, and let ϕɛ : → R be as above. Then there are
constants C, C ′ > 0 such that, for all u ∈ C ∞ (M0) supported in and h>0 small
(3)

96 GUILLARMOU and TZOU
enough, we have the estimate
C
ɛ u2 L2
(M0) + C′ − Im(〈∂τ u, u〉L2 (∂M0)) + 1
h
∂M0
|u| 2 ∂νϕɛdvg
≤e −ϕɛ/h¯∂e ϕɛ/h 2
uL2 (M0) , (4)
where ∂ν and ∂τ denote, respectively, the exterior pointing normal vector fields and
its rotation by an angle +π/2.
Proof
We use complex coordinates z = x +iy in the chart , where u is supported. Since the
Lebesgue measure dxdy is bounded below and above by dvg, g is conformal to |dz| 2 ,
and the boundary terms in (4) depend only on the conformal class, it then suffices to
prove the estimates with respect to dxdy and the Euclidean metric. We thus integrate
by parts with respect to dxdy and we have
4e −ϕɛ/h¯∂e ϕɛ/h
2
u = ∂x + i∂yϕɛ
u + i∂y +
h
∂xϕɛ
u
h
2
= ∂x + i∂yϕɛ
u
h
2
+ i∂y + ∂xϕɛ
u
h
2
+ 2
ϕɛ|u|
h
2 − 1
2 ∂xϕɛ.∂x|u| 2 − 1
2
∂yϕɛ.∂y|u|
2
+ 2
∂νϕɛ|u|
h ∂M0
2
− 2 ∂xRe(u).∂yIm(u)
M0
− ∂xIm(u).∂yRe(u)
(5)
= ∂x + i∂yϕɛ
u
h
2
+ i∂y + ∂xϕɛ
u
h
2
+ 1
ϕɛ|u|
h
2
+ 1
∂νϕɛ|u|
h ∂M0
2
+ 2 ∂τ Re(u).Im(u),
∂M0
where := −(∂ 2 x + ∂2 y ), where ∂ν is the exterior pointing normal vector field to the
boundary, and where ∂τ is the tangent vector field to the boundary (i.e., ∂ν rotated
with an angle π/2) for the Euclidean metric |dz| 2 .Then〈uϕɛ,u〉=(h/ɛ)(|dϕ0| 2 +
|dϕ1| 2 +···+|dϕN| 2 )|u| 2 ,sinceϕj are harmonic, so the proof follows from the fact
that |dϕ0| 2 +|dϕ1| 2 +···+|dϕN| 2 is uniformly bounded away from zero.
Themainsteptogofrom(4) to(3) is the following lemma, whose proof is a slight
modification of the one in [19, Proposition 5.3].

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 97
LEMMA 3.3
With the same assumptions as in Proposition 3.1, and if is either an interior chart
of (M0,g0) or a chart containing a whole boundary-connected component, then there
are positive constants C, C ′ such that for all ɛ>0 small, for all 0 0,
e −ϕɛ/h ϕɛ/h 2 C
e u ≥
ɛ
du 2 + 1
2
u|dϕɛ|
h2 + 2
h 〈∂xu, u∂xϕɛ〉+ 2
h 〈∂yu,
u∂yϕɛ〉
+ C ′
(B∂τA − A∂τ B)|∂νu| 2 + 1
h
∂M0
∂M0
|∂νu| 2 ∂νϕɛ
.
(7)

98 GUILLARMOU and TZOU
Using the fact that u is real-valued, that ϕ is harmonic, and that N
j=0 |dϕj| 2 is
uniformly bounded away from zero, we see that
2
h 〈∂xu, u∂xϕɛ〉+ 2
h 〈∂yu, u∂yϕɛ〉 = 1
h 〈u, uϕɛ〉 ≥ C
ɛ u2
for some C>0, and therefore, by possibly modifying C>0, wehave
e −ϕɛ/h ϕɛ/h 2 C
e u ≥ du
ɛ
2 + 1
h2 u|dϕɛ| 2 + C
ɛ u2
+ boundary terms. (8)
Now if the diameter of the support of u is chosen small (with size depending only on
|Hessϕ0|(p)) with a unique critical point p of ϕ0 inside, one can use integration by
parts and the fact that the critical point is nondegenerate to obtain
¯∂u 2 + 1
h2 u|∂ϕ0| 2 ≥ 1
∂¯z(u
h
2
)∂zϕ0dxdy
≥ 1
u
h
2 ∂2 z ϕ0
dxdy
≥ C′′
h u2
(9)
for some C ′′ > 0. Clearly, the same estimate holds trivially if does not contain a
critical point of ϕ0. Using a partition of unity (θj)j in and absorbing terms of the
form ||u¯∂θj|| 2 into the right-hand side, one obtains (9) for any function u supported
in and vanishing at the boundary. Thus, combining with (8), there are positive
constants C, C0,C1 such that, for h small enough, we have
C
du
ɛ
2 + 1
≥ C1
ɛ
h 2 u|dϕɛ| 2 + C
ɛ u2
du 2 + 1
h 2 u|dϕ0| 2 + 1
h u2
Combining now with (8)gives
≥ C
du
ɛ
2 + 1
.
h2 u|dϕ0| 2 − C0
ɛ
u2
2
e −ϕɛ/h
ϕɛ/h 2
C1
e u ≥ du
ɛ
2 + 1
h2 u|dϕ|2 + 1
h u2
+ boundary terms.
Let us now discuss the boundary terms in (7). If ϕj aretakensothat∂νϕj = 0 on Ɣ0,
then ∂νϕɛ = 0 on Ɣ0 and ∂νϕɛ = ∂νϕ + O(h/ɛ) on Ɣ, and thus
1
|∂νu|
h
2 |∂νϕɛ| ≤ C2
|∂νu|
h
2
∂M0
for some constant C2 > 0. We finally claim that B∂τ A − A∂τ B = 1 on ∂M0 ∩ .
Indeed, since the chart near a connected component can be taken to be an interior
neighborhood of the circle |z| =1 in C, one has A + iB = e −it , where t ∈ S 1
parameterize the boundary component, so that B∂τ A − A∂τ B = 1 since ∂τ = ∂t
Ɣ

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 99
for the Euclidean metric. Combining all these estimates and possibly modifying the
constants achieves the proof.
Proof of Proposition 3.1
Using triangular inequality and absorbing the term ||Vu|| 2 into the left-hand side of
(3), it suffices to prove (3) with g instead of g + V .Letv ∈ C ∞ 0 (M0); by Lemma
3.3, we have that there exist constants C, C ′ ,C1,C2 > 0 such that
C
1
ɛ h e−ϕɛ/h 2 1
v +
h2 e−ϕɛ/h 2 1
v|dϕ| +
h2 e−ϕɛ/hv|dϕɛ| 2 +e −ϕɛ/h
2
dv
≤
j
+e −ϕɛ/h ∂νv 2
Ɣ0
C ′
ɛ
≤ C1
≤ C2
1
h e−ϕɛ/h χjv 2 + 1
h 2 e−ϕɛ/h χjv|dϕ| 2 + 1
h 2 e−ϕɛ/h χjv|dϕɛ| 2
+e −ϕɛ/h
d(χjv) 2
+e −ϕɛ/h
∂νvƔ0
e −ϕɛ/h
g(χjv) 2 +e −ϕɛ/h
∂νv 2
Ɣ
j
−ϕɛ/h
e gv 2 +e −ϕɛ/h 2 −ϕɛ/h 2 −ϕɛ/h
v +e dv +e ∂νv 2
Ɣ ,
where (χj)j is a partition of unity associated to the complex charts j on M0. Since
constants on both sides are independent of ɛ and h, we can take ɛ small enough so that
C2e −ϕɛ/h v 2 + C2e −ϕɛ/h dv 2 can be absorbed to the left side. Now set v = e ϕɛ/h w
with w|∂M0 = 0. Then we have (for some new constant C>0)
1
h w2 + 1
h2 w|dϕ|2 + 1
h2 w|dϕɛ| 2 +dw 2 +∂νω 2
Ɣ0
≤ C e −ϕɛ/h
ge ϕɛ/h 2
w +∂νω 2
Ɣ
Finally, fix ɛ>0,setu := e 1 N ɛ j=0 |ϕj | 2
w, and use the fact that e 1 N ɛ j=0 |ϕj | 2
is independent
of h and is bounded uniformly away from zero and above; we then obtain the desired
estimate for 0 0, with phase a Carleman weight (here a Morse holomorphic function), and
such that the phase has a nondegenerate critical point at p, in order to apply the stationary
phase method. In general, complex geometric optic solutions are 1-parameter
families of solutions of the equation (g + V )u = 0 which are perturbations of

100 GUILLARMOU and TZOU
certain exponentially growing solutions (with respect to the parameter) of the free
equation gu = 0. They were introduced by Faddeev in [10] and are also called
Faddeev-type solutions. The oscillating part of the solutions is what will allow us to
recover information on the perturbation V .
In this section, the potential V has the regularity V ∈ C 1,α (M0) for some α>0.
Choose p ∈ int(M0) such that there exists a holomorphic function = ϕ +iψ which
is Morse on M0, C k in M0 for large k ∈ N, and such that ∂(p) = 0 and has
only finitely many critical points in M0. Furthermore, we ask that is purely real
on Ɣ0. By Proposition 2.3.1, such points p form a dense subset of M0. Given such a
holomorphic function, the purpose of this section is to construct solutions u on M0 of
( + V )u = 0 of the form
u = e /h (a + ha0 + r1) + e /h (a + ha0 + r1) + e ϕ/h r2 with u|Ɣ0 = 0 (10)
for h>0 small, where a is holomorphic and u ∈ C k (M0) for large k ∈ N, a0 ∈
H 2 (M0) is holomorphic, and moreover where a(p) = 0 and a vanishes to high order
at all other critical points p ′ ∈ M0 of . Furthermore, we ask that the holomorphic
function a is purely imaginary on Ɣ0. The existence of such a holomorphic function
is a consequence of Lemma 2.2.4. Given such a holomorphic function on M0, we
consider a compactly supported extension to M, still denoted a.
The remainder terms r1,r2 will be controlled as h → 0 and have particular
properties near the critical points of . More precisely, r2 will be a OL 2(h3/2 | log h|)
and r1 will be of the form hr12 + oL 2(h), where r12 is independent of h, which can
be used to obtain sufficient information from the stationary phase method in the
identification process.
4.1. Construction of r1
We will construct r1 to satisfy
e −/h (g + V )e /h (a + r1) = OL2(h| log h|)
and r1 = r11 + hr12. WeletGbe the Green operator of the Laplacian on the smooth
surface with boundary M0 with Dirichlet condition, so that gG = Id on L2 (M0).In
particular, this implies that ¯∂∂G = (i/2)⋆−1 , where ⋆−1 is the inverse of ⋆ mapping
functions to 2-forms. We extend a to be a compactly supported Ck function on M0,
and we will search for r1 ∈ H 2 (M0) satisfying ||r1||L2 = O(h) and
e −2iψ/h ∂e 2iψ/h r1 =−∂G(aV ) + ω + OH 1(h| log h|), (11)
where ω is a smooth holomorphic 1-form on M0. Indeed, using the fact that is
holomorphic, we have
e −/h ge /h =−2i ⋆¯∂e −/h ∂e /h =−2i ⋆¯∂e −(1/h)(− ¯) ∂e (1/h)(− ¯)
=−2i ⋆¯∂e −2iψ/h ∂e 2iψ/h ;

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 101
applying −2i ⋆¯∂ to (11), we obtain (note that ∂G(aV ) ∈ C 2,α (M0) by elliptic
regularity)
e −/h (g + V )e /h r1 =−aV + OL2(h| log h|).
We choose ω to be a smooth holomorphic 1-form on M0 such that at all critical
points p ′ of in M0, the form b := ∂G(aV ) − ω with value in T ∗
1,0M0 vanishes
to the highest possible order. Writing b = b(z)dz in local complex coordinates, b(z)
is C2+α by elliptic regularity, and we have −2i∂¯zb(z) = aV ; therefore, ∂z∂¯zb(p ′ ) =
∂2 ¯z b(p′ ) = 0 at each critical point p ′ = p by construction of the function a. Therefore,
we deduce that at each critical point p ′ = p, ∂G(aV ) has Taylor series expansion
2 j=0 cjzj + O(|z| 2+α ). That is, all the lower-order terms of the Taylor expansion of
∂G(aV ) around p ′ are polynomials of z only.
LEMMA 4.1.1
Let {p0,...,pN} be finitely many points on M0, and let θ be a C2,α section of T ∗
1,0M0. Then there exists a Ck holomorphic function f on M0 with k ∈ N large, such that f
vanishes to high order at the points {p1,...,pN} and ω = ∂f satisfies the following:
in complex local coordinates z near p0 , one has ∂ℓ z θ(p0) = ∂ℓ z ω(p0) for ℓ = 0, 1, 2,
where θ = θ(z) dz and ω = ω(z) dz.
Proof
This is a direct consequence of Lemma 2.2.4.
Applying this to the form ∂G(aV ) and using the observation made above, we can
construct a C k holomorphic form ω such that in local coordinates z centered at a critical
point p ′ of (i.e., p ′ ={z = 0} in this coordinate), for b =−∂G(aV )+ω = b(z) dz
we have
|∂ m
¯z ∂ℓ
z b(z)| =O(|z|2+α−ℓ−m ) for ℓ + m ≤ 2 if p ′ = p,
|b(z)| =O(|z|) if p ′ = p.
Now, we let χ1 ∈ C ∞ 0 (M0) be a cutoff function supported in a small neighborhood
Up of the critical point p and identically 1 near p, while χ ∈ C ∞ 0 (M0) is defined
similarly with χ = 1 on the support of χ1. We construct r1 = r11 + hr12 in two steps:
first, we construct r11 to solve (11) locally near the critical point p of , and second,
we construct the global correction term r12 away from p by using the extra vanishing
of b in (12) at the other critical points.
We define locally in complex coordinates centered at p and containing the support
of χ
r11 := χe −2iψ/h R(e 2iψ/h χ1b),
(12)

102 GUILLARMOU and TZOU
−1 where Rf (z) :=−(2πi) R2(1/¯z − ¯ξ)fd¯ξ ∧ dξ for f ∈ L∞ compactly supported
is the classical Cauchy-Riemann operator inverting locally ∂z (r11 is extended by zero
outside the neighborhood of p). The function r11 is in C3+α (M0), and we have
e −2iψ/h ∂(e 2iψ/h r11) = χ1
−∂G(aV ) + ω + η
with η := e −2iψ/h R(e 2iψ/h χ1b)∂χ. (13)
We then construct r12 by observing that b vanishes to order 2 + α at critical points of
other than p (from (12)), and ∂χ = 0 in a neighborhood of any critical point of ψ,
so we can find r12 satisfying
2ir12∂ψ = (1 − χ1)b.
This is possible since both ∂ψ and the right-hand side are valued in T ∗
1,0 M0, and∂ψ
has finitely many isolated zeroes on M0; indeed, r12 is then a function in C 2,α (M0 \ P )
where P := {p1,...,pN} is the set of critical points other than p, and it extends to a
C 1,α (M0) function which satisfies in local complex coordinates z near each pj
|∂ β
¯z ∂ γ
z r12(z)| ≤C|z − pj| 1+α−β−γ , β + γ ≤ 2,
where we used also the fact that ∂ψ can locally be considered as a holomorphic
function with a zero of order 1 at each pj. This implies that r1 ∈ H 2 (M0), andwe
have
e −2iψ/h ∂(e 2iψ/h r1) = b + h∂r12 + η =−∂G(aV ) − ω + h∂r12 + η.
Now the first error term ||∂r12||H 1 (M0) is bounded by
||∂r12||H 1 (1
− χ1)b(z)
(M0) ≤ C
∂zψ(z)
H 2 (Up)
≤ C
for some constant C, where we used the fact that (1 − χ1)b(z)/∂zψ(z) is in H 2 (Up)
and is independent of h. To deal with the η term, we need the following.
LEMMA 4.1.2
The estimates
hold true wherer12 solves 2ir12∂ψ = b.
||η||H 2 = O(| log h|),
||η||H 1 ≤ O(h| log h|),
||r1||L2 = O(h),
||r1 − hr12||L2 = o(h)

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 103
Proof
We start by observing that
||r1 − hr12||L2 =
||r1||L2 ≤
χe
−2iψ/h R(e 2iψ/h χ1b) − h χ1b
2i∂zψ
χe
−2iψ/h R(e 2iψ/h χ1b) − h χ1b
2i∂zψ
L 2 (Up)
L 2 (Up)
,
+ h||r12||L 2 (M0).
The first term is estimated in [19, Proposition 2.7] and it is a o(h), while ||r12||L 2
is independent of h. Now we are going to estimate the H 2 norms of η. Locally in
complex coordinates z centered at p (i.e., p ={z = 0}), we have
η(z) =−∂zχ(z)e −2iψ(z)/h
e 2iψ(ξ)/h 1 dξ1dξ2
χ1(ξ)b(ξ) ,
¯z − ¯ξ π
ξ = ξ1 +iξ2. (15)
C
Since b is C2,α in U, we decompose b(ξ) =〈∇b(0),ξ〉+b(ξ) using Taylor’s formula,
so we haveb(0) = ∂ξb(0) = 0 and we split the integral (15) with 〈∇b(0),ξ〉 andb(ξ).
Since the integrand with the 〈∇b(0),ξ〉 term is smooth and compactly supported in ξ
(recall that χ1 = 0 on the support of ∂zχ), we can apply stationary phase to get
∂zχ(z)e −2iψ(z)/h
e
C
2iψ(ξ)/h
1
¯z − ¯ξ
χ1(ξ)〈∇b(0),ξ〉 dξ1dξ2
π
≤ Ch 2
uniformly in z. Nowsetbz(ξ) = ∂zχ(z)χ1(ξ)b(ξ)/(z − ξ) which is C2,α in ξ and
smooth in z. Letθ∈ C∞ 0 ([0, 1)) be a cutoff function which is equal to 1 near zero,
and set θh(ξ) := θ(|ξ|/h). Then integrating by parts, we have
C
e 2iψ(ξ)/hbz(ξ)dξ1dξ2 = h 2
− h
e
supp(χ1)
2iψ(ξ)/h ∂¯ξ
supp(χ1)
e 2iψ(ξ)/h θh(ξ)∂ξ
(14)
1 − θh(ξ)
2i∂¯ξψ ∂ξ
bz(ξ)
dξ1dξ2
2i∂ξψ
bz(ξ)
dξ1dξ2. (16)
2i∂ξψ
Using polar coordinates with the fact that bz(0) = 0, it is easy to check that the
second term in (16) is bounded uniformly in z by Ch 2 . To deal with the first term,
we use bz(0) = ∂ξ bz(0) = ∂¯ξ bz(0) = 0 and a straightforward computation in polar
coordinates shows that the first term of (16) is bounded uniformly in z by Ch 2 | log(h)|.
We conclude that
||η||L 2 ≤ C||η||L ∞ ≤ Ch2 | log h|.
It is also direct to see that the same estimates hold with a loss of h −2 for any derivatives
in z, ¯z of order less or equal to 2, since they only hit the χ(z) factor, the (¯z − ¯ξ) −1

104 GUILLARMOU and TZOU
factor, or the oscillating term e −2iψ(z)/h . So we deduce that
||η||H 2 = O(| log h|)
and this ends the proof.
We summarize the result of this section with the following.
LEMMA 4.1.3
Let k ∈ N be large, and let ∈ C k (M0) be a holomorphic function on M0 which is
Morse in M0 with a critical point at p ∈ int(M0). Leta ∈ C k (M0) be a holomorphic
function on M0 vanishing to high order at every critical point of other than p.Then
there exists r1 ∈ H 2 (M0) such that ||r1||L2 = O(h) and
e −/h ( + V )e /h (a + r1) = OL2(h| log h|).
4.2. Construction of a0
We have constructed the correction term r1 which solves the Schrödinger equation to
order h as stated in Lemma 4.1.3. In this section, we construct a holomorphic function
a0 which annihilates the boundary value of the solution on Ɣ0. In particular, we have
the following.
LEMMA 4.2.1
There exists a holomorphic function a0 ∈ H 2 (M0) independent of h such that
and
e −/h ( + V )e /h (a + r1 + ha0) = OL2(h| log h|)
[e /h (a + r1 + ha0) + e /h (a + r1 + ha0)]|Ɣ0 = 0.
Proof
First, notice that h −1 r1|∂M0 = r12|∂M0 ∈ H 3/2 (∂M0) is independent of h. Since is
purely real on Ɣ0 and a is purely imaginary on Ɣ0, we see that this lemma amounts to
constructing a holomorphic function a0 ∈ H 2 (M0) with the boundary condition
Re(r12) + Re(a0) = 0 on Ɣ0.
To construct a0, it suffices to use item (ii) in Corollary 2.2.3.
4.3. Construction of r2
The goal of this section is to complete the construction of the complex geometric optic
solutions by the following proposition.

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 105
PROPOSITION 4.3.1
There exist solutions to ( + V )u = 0 with boundary condition u|Ɣ0 = 0 of the
form (10) with r1, a0 constructed in the previous sections and r2 satisfying r2L2 =
O(h3/2 | log h|).
This is a consequence of the following lemma (which follows from the Carleman
estimate obtained above).
LEMMA 4.3.2
If V ∈ L ∞ (M0) and f ∈ L 2 (M0), then for all h>0 small enough, there exists a
solution v ∈ L 2 to the boundary-value problem
satisfying the estimate
e ϕ/h (g + V )e −ϕ/h v = f, v|Ɣ0 = 0,
vL 2 ≤ Ch1/2 f L 2.
Proof
The proof is identical to the proof of [19, Proposition 2.2]; we repeat the argument
here for the convenience of the reader. Define for all h>0 the vector space A :=
{u ∈ H 1 0 (M0); (g + V )u ∈ L2 (M0), ∂νu |Ɣ= 0} equipped with the scalar product
(u, w)A := e −2ϕ/h (gu + Vu)(gw + Vw)dvg.
M0
Since ψ is constant along Ɣ0 and since ϕ + iψ is holomorphic, then ∂νϕ = 0 on Ɣ0.
Therefore, we may apply the Carleman estimate of Proposition 3.1 to the weight ϕ
to assert that the space A is a Hilbert space equipped with the scalar product above.
By using the same estimate, the linear functional L : w →
M0 e−ϕ/hfwdvg on A
is continuous and its norm is bounded by h1/2 ||f ||L2. By the Riesz theorem, there is
an element u ∈ A such that (u, .)A = L and with norm bounded by the norm of L.
It remains to take v := e−ϕ/h (gu + Vu), which solves (g + V )e−ϕ/hv = e−ϕ/hf and, in addition, satisfies the desired norm estimate. Furthermore, since
e −ϕ/h
v(g + V )w dvg = e −ϕ/h fwdvg
M0
for all w ∈ A, by Green’s theorem we have
e −ϕ/h
v∂νw dvg = 0 =
∂M0
Ɣ0
M0
e −ϕ/h v∂νw dvg
for all w ∈ A. Thisimpliesthatv = 0 on Ɣ0.

106 GUILLARMOU and TZOU
Proof of Proposition 4.1
We note that
( + V ) e /h (a + r1 + ha0) + e/h (a + r1 + ha0) + e ϕ/h
r2 = 0
if and only if
e −ϕ/h ( + V )e ϕ/h r2 =−e −ϕ/h ( + V ) e /h (a + r1 + ha0) + e /h (a + r1 + ha0) .
By Lemma 4.2.1, the right-hand side of the above equation is OL2(h| log h|). Therefore,
using Lemma 4.3.2, one can find such r2 which satisfies
r2L 2 ≤ Ch3/2 | log h|, r2 |Ɣ0= 0.
Since the ansatz e /h (a + r1 + ha0) + e /h (a + r1 + ha0) is arranged to vanish on
Ɣ0, the solution
u = e /h (a + r1 + ha0) + e /h (a + r1 + ha0) + e ϕ/h r2
vanishes on Ɣ0 as well.
5. Identifying the potential
We now assume that V1,V2 ∈ C 1,α (M0) are two potentials, with α>0, such that
the respective Cauchy data spaces C Ɣ 1 , C Ɣ 2 for the operators g + V1 and g + V2
on Ɣ ⊂ ∂M0 are equal. We may also assume that V1 = V2 on Ɣ by boundary
identification, a fact which is proved below in the appendix. Let Ɣ0 = ∂M0 \ Ɣ be the
complement of Ɣ in ∂M0, and possibly by taking Ɣ slightly smaller, we may assume
that Ɣ0 contains an open set. Let p ∈ M0 be an interior point of M0 such that, using
Proposition 2.3.1, we can choose a holomorphic Morse function = ϕ + iψ on M0
with purely real on Ɣ0, C k in M0 for some large k ∈ N, with a critical point at p.
Note that Proposition 2.3.1 states that we can choose such that none of its critical
points on the boundary are degenerate and such that critical points do not accumulate
on the boundary.
PROPOSITION 5.1
If the Cauchy data spaces agree, that is, if C Ɣ 1 = C Ɣ 2 ,thenV1(p) = V2(p).
Proof
Let a be a holomorphic function on M0 which is purely imaginary on Ɣ0 with a(p) = 0
and a(p ′ ) = 0 to large order for all other critical points p ′ of . The existence of a is

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 107
ensured by Lemma 2.2.4.Letu1 and u2 be H 2 solutions on M0 to
(g + Vj)uj = 0
constructed in Section 4 with = φ + iψ for Carleman weight for u1 and − for
u2, and thus of the form
u1 = e /h (a + ha0 + r1) + e /h (a + ha0 + r1) + e ϕ/h r2,
u2 = e −/h (a + hb0 + s1) + e −/h (a + hb0 + s1) + e −ϕ/h s2,
and with boundary value uj|∂M0 = fj, where fj vanishes on Ɣ0. By Green’s formula,
we can write
M0
u1(V1 − V2)u2 dvg =−
=−
M0
∂M0
(gu1.u2 − u1.gu2)dvg
(∂νu1.f2 − f1.∂νu2)dvg.
Since the Cauchy data for g + V1 agrees on Ɣ with that of g + V2, there exists a
solution v of the boundary-value problem
(g + V2)v = 0, v|∂M0 = f1,
satisfying ∂νv = ∂νu1 on Ɣ. Sincefj = 0 on Ɣ0, thisimpliesthat
u1(V1 − V2)u2 dvg =− (gu1.u2 − u1.gu2)dvg
M0
=−
=−
=−
M0
∂M0
∂M0
M0
(∂νu1.f2 − f1.∂νu2)dvg
(∂νv.f2 − v.∂νu2)dvg
(gv.u2 − v.gu2)dvg = 0 (17)
since g + V2 annihilates both v and u2. We substitute in the full expansion for u1
and u2, and setting V := V1 − V2 and using the estimates in Lemmas 4.1.2, 4.3.1,and
4.2.1, we obtain
0 = I1 + I2 + o(h), (18)

108 GUILLARMOU and TZOU
where
I1 =
M0
I2 = 2h
V (a 2 + a 2
)dvg + 2
M0
V Re ae 2iψ/h
s1
h
+ e −2iψ/h
a0 + r1
h
M0
V |a| 2 Re(e 2iψ/h )dvg, (19)
+ b0
+ b0 + a0 + s1 + r1
h
dvg. (20)
Remark 5.2
We observe from the last identity in Lemma 4.1.2 that r1/h in the expression I2 can
be replaced by the term r12 satisfying 2ir12∂ψ = b up to an error which can go in
the o(h) in (18), and similarly for the term s1/h, which can be replaced by a terms12
independent of h.
We apply the stationary phase to these two terms in Lemmas 5.3 and 5.4.
LEMMA 5.3
The estimate
I2 = 2h
M0
V Re
a(b0 + a0 +s12 +r12)dvg + o(h)
holds true where r12, s12,r12 ands12 are independent of h.
Proof
We start with the following.
LEMMA 5.4
Let f ∈ L 1 (M0). Thenash → 0,
M0
e 2iψ/h f dvg = o(1).
Proof
Since C k (M0) is dense in L 1 (M0) for all k ∈ N, it suffices to prove the lemma for
f ∈ C k (M0).Letɛ>0 be small, and choose a cutoff function χ which is identically
equal to 1 on the boundary such that
M0
χ|f | dvg ≤ ɛ.

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 109
Then, splitting the integral and using stationary phase for the 1 − χ term, we obtain
e 2iψ/h
f dvg
≤ (1 − χ)e 2iψ/h
f dvg
+ χe 2iψ/h
f dvg
≤ ɛ + Oɛ(h),
M0
M0
which concludes the proof by taking h small enough depending on ɛ.
The proof of Lemma 5.3 is a direct consequence of Lemma 5.4 and Remark
5.2.1.
The second lemma will be proved at the end of this section.
LEMMA 5.5
The estimate
I1 =
M0
V (a 2 + a 2 )dvg + hCpV (p)|a(p)| 2 Re(e 2iψ(p)/h ) + o(h)
holds true with Cp = 0 and is independent of h.
With Lemmas 5.3 and 5.4, we can write (18) as
0 = V (a 2 + a 2 )dvg + O(h)
M0
and thus we can conclude that
0 =
M0
M0
V (a 2 + a 2 )dvg.
Therefore, (18) becomes, with Lemma 5.3,
0 = CpV (p)|a(p)| 2 Re(e 2iψ(p)/h
) + 2 V Re a(b0 + ˜s12 + a0 + ˜r12) dvg + o(1).
M0
Since ψ(p) = 0, we may choose a sequence of hj → 0 such that Re(e 2iψ(p)/hj ) = 1
and another sequence ˜hj → 0 such that Re(e 2iψ(p)/˜hj ) =−1 for all j. Adding the
expansion with h = hj and h = ˜hj, we deduce that
0 = 2CpV (p)|a(p)| 2 + o(1)
as j →∞, and since Cp = 0, a(p) = 0, we conclude that V (p) = 0. The set
of p ∈ M0 for which we can conclude this is dense in M0, by Proposition 2.3.1.
Therefore, we can conclude that V (p) = 0 for all p ∈ M0.
We now prove Lemma 5.5.

110 GUILLARMOU and TZOU
Proof of Lemma 5.5
Let χ be a smooth cutoff function on M0 which is identically 1 everywhere except
outside a small ball containing p and no other critical point of , andletχ = 0 near
p. We split the oscillatory integral into two parts:
(e 2iψ/h + e −2iψ/h )V |a| 2
dvg = χ(e 2iψ/h + e −2iψ/h )V |a| 2 dvg
M0
M0
+
M0
(1 − χ)(e 2iψ/h + e −2iψ/h )V |a| 2 dvg.
The phase ψ has nondegenerate critical points. Therefore, a standard application of
the stationary phase at p gives
(1 − χ)(e 2iψ/h + e −2iψ/h )V (p)|a| 2 dvg = hCp|a(p)| 2 V (p)Re(e 2iψ(p)/h ) + o(h),
M0
where Cp is a nonzero number which depends on the Hessian of ψ at the point p.
Define the potential V (·) := V (·) − V (p) ∈ C1,α (M0). Then we show that
(1 − χ)(e 2iψ/h + e −2iψ/h )V |a| 2 dvg = o(h). (21)
M0
Indeed, first by integration by parts and using gψ = 0, one has
(1 − χ)(e 2iψ/h + e −2iψ/h )V |a| 2 dvg
M0
= h
2i
= h
2i
M0
M0
〈d(e 2iψ/h − e −2iψ/h (1 − χ)|a|2
),dψ〉V
|dψ| 2
dvg
(e 2iψ/h − e −2iψ/h
(1 − χ)|a| 2V
) d
|dψ| 2
,dψ dvg.
But we can see that 〈d((1 − χ)|a| 2V/| dψ| 2 ),dψ〉∈L1 (M0): this follows directly
from the fact that V is in the Hölder space C1,α (M0) and V (p) = 0, and from the
nondegeneracy of Hess(ψ). It then suffices to use Lemma 5.4 to conclude that (21)
holds. Using a similar argument, we now show that
χ(e 2iψ/h + e −2iψ/h )V |a| 2 dvg = o(h).
M0

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 111
Indeed, since a vanishes to large order at all boundary critical points of ψ, we may
write
χ(e 2iψ/h + e −2iψ/h )V |a| 2 dvg
M0
= h
〈d(e
2i M0
2iψ/h − e −2iψ/h ),dψ〉V χ|a|2
dvg
|dψ| 2
=− h
(e
2i M0
2iψ/h − e −2iψ/h
)divg V χ|a|2
|dψ| 2 ∇g
ψ dvg
+ h
(e
2i
2iψ/h − e −2iψ/h )V |a|2
|dψ| 2 ∂νψ dvg.
∂M0
For the interior integral, we use Lemma 5.4 to conclude that
− h
(e
2i M0
2iψ/h − e −2iψ/h
)divg V χ|a|2
|dψ| 2 ∇g
ψ dvg = o(h),
and for the boundary integral we write ∂M0 = Ɣ0 ∪ Ɣ and observe that on Ɣ0, ψ = 0,
so (e2iψ/h−e−2iψ/h ) = 0, while on Ɣ we have V = 0 from the boundary determination
proved in Proposition A.1 of the appendix. Therefore,
χ(e 2iψ/h + e −2iψ/h )V |a| 2 dvg = o(h),
M0
and the proof is complete.
6. Inverse scattering
We first obtain, as a trivial consequence of Theorem 1.1, a result about inverse
scattering for asymptotically hyperbolic (AH) surfaces. Recall that an AH surface
is an open complete Riemannian surface (X, g) such that X is the interior of a
smooth compact surface with boundary ¯X, and for any smooth boundary-defining
function x of ∂ ¯X, ¯g := x 2 g extends as a smooth metric to ¯X, with curvature tending
to −1 at ∂ ¯X. IfV ∈ C ∞ ( ¯X) and if V = O(x 2 ), then we can define a scattering
map as follows (see [20], [13] or[14]). First, the L 2 kernel kerL 2(g + V ) is
a finite-dimensional subspace of xC ∞ ( ¯X) and in one-to-one correspondence with
E := {(∂xψ)| ∂ ¯X; ψ ∈ kerL 2(g + V )}, where ∂x := ∇ ¯g x is the normal vector field to
∂ ¯X for ¯g. Then, for f ∈ C ∞ (∂ ¯X), there exists a function u ∈ C ∞ ( ¯X), unique modulo
kerL 2(g + V ), such that (g + V )u = 0 and u| ∂ ¯X = f . Then one can see that the
scattering map S : C ∞ (∂ ¯X) → C ∞ (∂ ¯X)/E is defined by Sf := ∂xu| ∂ ¯X. We thus
obtain the following.

112 GUILLARMOU and TZOU
COROLLARY 6.1
Let (X, g) be an asymptotically hyperbolic manifold, and let V1,V2 ∈ x 2 C ∞ ( ¯X) be
two potentials and Ɣ ⊂ ∂ ¯X an open subset of the conformal boundary. Assume that
∂xu ∂ ¯X ; u ∈ kerL 2(g + V1) = ∂xu ∂ ¯X ; u ∈ kerL 2(g + V2) ,
and let Sj be the scattering map for the operator g + Vj for j = 1, 2.IfS1f = S2f
on Ɣ for all f ∈ C ∞ 0 (Ɣ), thenV1 = V2.
Proof
Let x be a smooth boundary-defining function of ∂ ¯X,let¯g = x 2 g be the compactified
metric, and define ¯Vj := Vj/x 2 ∈ C ∞ ( ¯X). By conformal invariance of the Laplacian
in dimension 2, one has
g + Vj = x 2 (¯g + ¯Vj),
and so if kerL 2(g + V1) = kerL 2(g + V2) and if S1 = S2 on Ɣ, then the Cauchy
data spaces C Ɣ i for the operator ¯g + ¯Vj are the same. Then it suffices to apply the
result in Theorem 1.1.
Next we consider the asymptotically Euclidean scattering at zero frequency. An
asymptotically Euclidean surface is a noncompact Riemann surface (X, g) which
compactifies into a smooth surface ¯X with boundary such that the metric, in a collar
neighborhood (0,ɛ)x × ∂ ¯X of the boundary, is of the form
g = dx2 h(x)
+
x4 x
where h(x) is a smooth 1-parameter family of metrics on ∂ ¯X such that h(0) = dθ2 S1 is
the metric. Notice that using the coordinates r := 1/x, g is asymptotic to dr2 +r 2dθ2 S1 near r →∞. A particular case is given by the surfaces with Euclidean ends, that
is, ends isometric to R2 \ B(0,R), where B(0,R) ={z ∈ R2 ; |z| ≥R}. Note that
g is conformal to an asymptotically cylindrical metric (or “b-metric” in the sense of
Melrose [27])
2 ,
gb := x 2 g = dx2
+ h(x)
x2 and that the Laplacian satisfies g = x2gb . Each end of X is of the form (0,ɛ)x ×S 1 θ ,
and the operator gb has the expression in the ends
gb =−(x∂x) 2 + ∂ ¯X + xP(x,θ; x∂x,∂θ)

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 113
for some smooth differential operator P (x,θ; x∂x,∂θ) in the vector fields x∂x,∂θ
down to x = 0. Let us define Vb := x −2 V , which is compactly supported, and
H 2m
b := u ∈ L 2 (X, dvolgb); m
gb u ∈ L2 (X, dvolgb) , m ∈ N0.
We also define the following spaces for α ∈ R:
Fα := ker x α H 2 b (gb + Vb).
Since the eigenvalues of S 1 are {j 2 ; j ∈ N0}, the relative index theorem of Melrose
[27, Section 6.2] shows that gb + Vb is Fredholm from xαH 2 b to xαH 0 b if α /∈ Z.
Moreover, from [27, Section 2.2.4], we have that any solution of (gb + Vb)u = 0 in
xαH 2 b has an asymptotic expansion of the form
u ∼
ℓj
j>α,j∈Z ℓ=0
x j (log x) ℓ uj,ℓ(θ) as x → 0
for some sequence (ℓj)j of nonnegative integers and some smooth function uj,ℓ on
S 1 . In particular, it is easy to check that kerL 2 (X,dvolg)(g + V ) = F1+ɛ for ɛ ∈ (0, 1).
THEOREM 6.2
Let (X, g) be an asymptotically Euclidean surface, let V1,V2 be two compactly supported
smooth potentials, and let x be a boundary-defining function. Let ɛ ∈ (0, 1),
and assume that, for any j ∈ Z and any function ψ ∈ ker x j−ɛ H 2 b (g + V1), thereisa
ϕ ∈ ker x j−ɛ H 2 b (g + V2) such that ψ − ϕ = O(x ∞ ), and conversely. Then V1 = V2.
Proof
The idea is to reduce the problem to the compact case. First, we notice that by unique
continuation, ψ = ϕ where V1 = V2 = 0. Now it remains to prove that, if Rη denotes
the restriction of smooth functions on X to {x ≥ η} and if V is a smooth compactly
supported potential in {x ≥ η}, then the set ∞ j=0 Rη(F−j−ɛ) is dense in the set NV
of H 2 ({x ≥ η}) solutions of (g + V )u = 0. The proof is well known for positive
frequency scattering (see, e.g., [28, Lemma 3.2]). Here it is very similar, so we do not
give details. The main argument is to show that it converges in the L2 sense and then
uses elliptic regularity; the L2 convergence can be shown as follows. Let f ∈ NV
such that
∞
fψdvolg = 0, ∀ ψ ∈ F−j−ɛ.
x≥η
Then we want to show that f = 0. By[27, Proposition 5.64], there exists k ∈ N
and a generalized right inverse Gb for Pb = gb + Vb (here, as before, x 2 Vb = V )
j=0

114 GUILLARMOU and TZOU
in x−k−ɛ H 2 b such that PbGb = Id. This holds in x−k−ɛ H 2 b for k large enough since
the cokernel of Pb on this space becomes zero for k large. Let ω = Gbf so that
(gb + Vb)ω = f ; in particular, this function is zero in {x n,
while Brown [5] studied the case of Lipschitz domains with a continuous conductivity.
Since the result in our setting is not explicitly written down but is certainly known
from specialists, we provide a short proof with few details using the approach of [5].
We prove the following.

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 115
PROPOSITION A.1
Let Ɣ ⊂ ∂M0 be a nonempty open subset of the boundary. If V1,V2 ∈ C0,α (M0) for
some α>0 and if their associated Cauchy data spaces C Ɣ 1 , C Ɣ 2 defined in (1) are
equal, then V1|Ɣ = V2|Ɣ.
The key to proving this proposition is the existence of solutions to (g + Vi)u = 0,
which concentrate near a point p ∈ Ɣ. First, we need a solvability result for the
equation (g + Vi)u = f , which is an easy consequence of the Carleman estimate
of Proposition 3.1, and follows the method of Salo and Tzou [31, Section 6]. If we
fix h>0 small and take ϕ = 1 in the Carleman estimate of Proposition 3.1, we
easily obtain that there is a constant C such that for all functions in H 2 (M0) satisfying
u|∂M0 = 0,
u 2
2
H 2 +∂νuL2 (Ɣ0)
2
2
≤ C(( + Vi)uL2 +∂νuL2 (Ɣ) ). (A.1)
As a consequence, we deduce the following solvability result. Let
B := w ∈ H 2 (M0) ∩ H 1
0 (M0) | ∂νw|Ɣ = 0
be the closed subspace of H 2 (M0) under the H 2 norm, and let B ∗ be its dual space.
Then we have the following.
COROLLARY A.2
Let i = 1, 2. Then for all f ∈ L 2 (M0) there exists u ∈ H 2 (M0) solving the equation
(g + Vi)u = f
with boundary condition u|Ɣ0 = 0, and uL2 ≤ Cf B∗. Proof
Set A := {w ∈ H 1 0 (M0) | (g + Vi)w ∈ L2 ,∂νw|Ɣ = 0} equipped with the inner
product
(v, w)A := (g + Vi)v(g + Vi)w dvg.
M0
Thanks to (A.1), A is a Hilbert space and A = B. For each f ∈ L 2 (M0), let us define
the linear functional on B:
By (A.1), we have, for all w ∈ B,
Lf : w ↦→
M0
wf dvg.
|Lf (w)| ≤f B ∗wB ≤f B ∗wA.

116 GUILLARMOU and TZOU
Therefore, by the Riesz theorem, there exists vf ∈ A such that
(g + V )vf (g + Vi)w dvg = wf dvg
M0
for all w ∈ B. Furthermore, (g + Vi)vf L2 ≤fB∗. Setting u := (g + Vi)vf ,
we have (g + Vi)u = f and uL2 ≤fB∗. To obtain the boundary condition for
u, observe that since
M0
u(g + Vi)wdvg =
for all w ∈ B, then by Green’s theorem,
u∂νw dvg = 0 =
M0
Ɣ0
M0
M0
fwdvg
u∂νw dvg
for all w ∈ B. Thisimpliesthatu = 0 on Ɣ0.
Clearly, it suffices to assume that Ɣ is a small piece of the boundary which is contained
in a single coordinate chart with complex coordinates z = x + iy, where |z| ≤1,
Im(z) > 0, and the boundary is given by {y = 0}. Moreover, the metric is of the
form e 2ρ |dz| 2 for some smooth function ρ.Letp ∈ Ɣ, and possibly by translating the
coordinates, we can assume that p ={z = 0}. Letη ∈ C ∞ (M0) be a cutoff function
supported in a small neighborhood of p. Forh>0 small, we define the smooth
function vh ∈ C ∞ (M0) supported near p via the coordinate chart Z = (x,y) ∈ R 2 by
vh(Z) := η(Z/ √ h)e (1/h)α·Z , (A.2)
where α := (i, −1) ∈ C 2 is chosen such that α · α = 0 and the dot product · means
the canonical symmetric C-bilinear form on C 2 . Thus we get (∂ 2 x + ∂2 y )eα·Z = 0 and
thus ge α·Z = 0 by conformal covariance of the Laplacian. Therefore, we have in
local coordinates
gvh(Z) = 1
h e(1/h)α·Z (gη)
Z√h
+ 2
Z√h
e(1/h)α·Z dη ,α· dZ . (A.3)
h3/2 g
LEMMA A.3
If V ∈ C 0,α (M0) for q>2, then there exists a solution uh ∈ H 2 to (g + V )u = 0
of the form
uh = vh + Rh,
with vh defined in (A.2) and RhL 2 ≤ Ch5/4 , satisfying supp(Rh|∂M0) ⊂ Ɣ.

CALDERÓN INVERSE PROBLEM ON RIEMANN SURFACES 117
Proof of Lemma A.3
We need to find Rh satisfying RhL 2 ≤ Ch5/4 and solving
(g + V )Rh =−(g + V )vh =: Mh.
Thanks to Corollary A.2, it suffices to show that MhB ∗ ≤ Ch5/4 . Thus, let w ∈ B;
then we have, by (A.3),
wMh dvg = I1 + I2 + I3,
where
I1 : =
I2 : = 1
h
M0
|Z|≤ √ h
I3 : = 2
h 3/2
wVivhe 2ρ dZ,
|Z|≤ √ we
h
(1/h)α·Z χ1
|Z|≤ √ wχ2
h
Z√h
e 2ρ dZ,
Z√h
e (1/h)α·Z dZ,
and χ1 = gη, χ2 = i∂xη − ∂yη. In the above equation, the term I3 has the worst
growth when h → 0. We analyze its behavior, and the preceding terms can be treated
in similar fashion. One has
I3 =−h 1/2
=−h 1/2
|Z|≤ √ wχ2
h
|Z|≤ √ (∂x − i∂y)
h
2
Z√h
(∂x − i∂y) 2 e (1/h)α·Z dZ
Z√h
wχ2 e (1/h)α·x dZ.
Notice that the boundary term in the integration by parts vanishes because w ∈ H 1 0
and ∂νw|∂M0 vanishes on the support of η. The term (∂x − i∂y) 2 (wχ2(Z/ √ h)) has
derivatives hitting both χ2(Z/ √ h) and w. The worst growth in h would occur when
both derivatives hit χ2(Z/ √ h), in which case a h −1 factor would come out. Combined
with the h 1/2 term in front of the integral this gives a total of a h −1/2 in front. By this
observation we have improved the growth from h −3/2 to h −1/2 . Repeating this line of argument
and using the Cauchy-Schwarz inequality, we can see that |I3| ≤Ch 5/4 wH 2
(an elementary computation shows that functions of the form χ(Z/ √ h)e (1/h)α·Z have
L 2 norm bounded by Ch 3/4 ). Therefore, (1/h 3/2 )χ2(Z/ √ h)e (1/h)α·Z B ′ ≤ Ch5/4 ,
and we are done.

118 GUILLARMOU and TZOU
Proof of Proposition A.1
It suffices to plug the solutions u1 h ,u2 h from Proposition A.3 into the boundary integral
identity (17). A simple calculation using the fact that V1 − V2 is in C0,γ (M0) yields
0 = u 1
h (V1 − V2)u 2
h dvg = Ch 3/2 V1(p) − V2(p) + o(h 3/2 ),
M0
and we are done.
Acknowledgments. This work started during a summer evening in Pisa thanks to the
hospitality of M. Mazzucchelli and A. G. Lecuona. We thank Sam Lisi, Rafe Mazzeo,
Mikko Salo, and Eleny Ionel for pointing out very helpful references. The first author
thanks the Mathematical Sciences Research Institute and the organizers of the 2008
program “Analysis on Singular Spaces” for support during part of this project. Part of
this work was also done while the first author was at the Institute for Advanced Study
in Princeton, and he acknowledges the hospitality extended to him.
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Guillarmou
École Normale Supérieure, UMR CNRS 8553, F 75230 Paris CEDEX 05, France;
cguillar@dma.ens.fr
Tzou
Department of Mathematics, Stanford University, Stanford, California 94305, USA;
ltzou@math.stanford.edu

QUANTITATIVE VERSION OF THE OPPENHEIM
CONJECTURE FOR INHOMOGENEOUS
QUADRATIC FORMS
GREGORY MARGULIS and AMIR MOHAMMADI
Abstract
We prove a quantitative version of the Oppenheim conjecture for inhomogeneous
quadratic forms. We also give an application to eigenvalue spacing on flat 2-tori with
Aharonov-Bohm flux.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2. Passage to space of inhomogeneous lattices . . . . . . . . . . . . . . . 128
3. The case where p ≥ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4. The case of signature (2, 2) . . . . . . . . . . . . . . . . . . . . . . . . 132
5. Contribution from quasi-null subspaces . . . . . . . . . . . . . . . . . 136
6. Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7. Proof of Theorem 1.10 . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
A. Equidistribution of spherical averages . . . . . . . . . . . . . . . . . . . 153
B. Number of quasi-null subspaces . . . . . . . . . . . . . . . . . . . . . . 157
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
1. Introduction
Let Q be a nondegenerate indefinite quadratic form on R n . Let ξ ∈ R n beavector,
and define the (inhomogeneous) quadratic form Qξ by
Qξ(x) = Q(x + ξ) for all x ∈ R n . (1)
We will refer to Q = Q0 as the homogeneous part of Qξ. We say that Qξ has signature
(p, q) if Q does. Recall that a quadratic form Qξ is called irrational if it is not a scalar
DUKE MATHEMATICAL JOURNAL
Vol. 158, No. 1, c○ 2011 DOI 10.1215/00127094-1276319
Received 4 March 2010. Revision received 22 July 2010.
2010 Mathematics Subject Classification. Primary 11E20; Secondary 58J50.
Margulis’s work partially supported by National Science Foundation grant DMS-0801195.
121

122 MARGULIS and MOHAMMADI
multiple of a form with rational coefficients. In other words, Qξ is irrational if either Q
is irrational as a homogeneous form, or, if Q is a rational form, then ξ is an irrational
vector.
Let ν be a continuous function on the sphere {v ∈ R n : v =1}. Define
={v ∈ R n : v

INHOMOGENEOUS QUADRATIC FORMS 123
Recall from [EMM2]thatifQ is irrational of signature (2, 2), then it has at most four
rational null subspaces. Let
ÑQ,(a,b,T ) = #
x ∈ Z n :
Eskin, Margulis, and Mozes proved the following.
x is not in a null subspace of Q
. (6)
x ∈ T and a

124 MARGULIS and MOHAMMADI
As in [EMM1], we also have the following uniform version of Theorem 1.5. Let
I(p, q) denote the space of inhomogeneous quadratic forms whose homogeneous
parts are quadratic forms of signature (p, q) and discriminant ±1.
THEOREM 1.6
Let D be a compact subset of I(p, q) with p ≥ 3 and q ≥ 1, and let n = p + q.
Then for every interval (a,b) and every θ>0, there exists a finite subset P of D
such that each Qξ ∈ P is a rational form and, for every compact subset F ⊂ D \ P ,
there exists T0 such that, for all Qξ ∈ F and T ≥ T0, we have
(1 − θ)λQ,(b − a)T n−2 ≤ NQ,ξ,(a,b,T ) ≤ (1 + θ)λQ,(b − a)T n−2 , (10)
where λQ, is as in (3).
The proofs of the above theorems are straightforward inhomogeneous versions of the
arguments and ideas developed in [DM] and[EMM1].
As we mentioned before, Theorem 1.5 fails in signature (2, 2) and (2, 1). In this,
paper, we prove an inhomogeneous version of Theorem 1.3. Indeed,asin[EMM2],
one needs to assume certain Diophantine conditions on the quadratic form. Using
similar ideas, we also give a partial result in the (2, 1) case (see Theorem 1.10 below).
We start with the following definitions.
Definition 1.7
Let κ>0. A vector ξ = (ξ1,...,ξn) ∈ R n is called κ-Diophantine if there exist C =
C(ξ) > 0 such that, for all 0

INHOMOGENEOUS QUADRATIC FORMS 125
then there are at most four subspaces L for which the above can hold. Let
ÑQ,ξ,(a,b,T ) = #
x ∈ Z n :
x is not in an exceptional subspace of Qξ
x ∈ T and a

126 MARGULIS and MOHAMMADI
planes” interfere with each other. This fact prevents us from successfully using the
ideas in [EMM2] when the signature is (2, 1). The absence of such an asymptotic
in the homogeneous case with explicit Diophantine conditions is responsible for the
“extra” assumption (i) in Theorem 1.10. Indeed, using this assumption, we are able to
reduce the study to an investigation of the contribution from null vectors. We are then
able to impose and utilize an explicit Diophantine condition on ξ (see Section 7 for
details).
Let us also mention that the Diophantine conditions in Theorems 1.10 and 1.9
are necessary. Similar to [EMM1, Theorem 2.2], [Mark, Theorem 1.3], and [Sar,
Theorem 2], it is possible to construct sets of second Baire category of quadratic
forms for which the above fail. To be more explicit, it is shown in [EMM1], using
category arguments, that there exists a dense set of second Baire category γ such
that NQ,(1/8, 2,Ti) >Ti(log Ti) 1−ε for an infinite sequence Ti, where Qγ (x) =
x2 1 + x2 2 − γ 2 (x2 3 + x2 4 ). Nowifwetakeanyξ∈ Z4 , then the same holds for Q γ
ξ .
This already gives the examples we wanted. However, these examples are in a sense
“degenerate" since ξ ∈ Z4 . We will reproduce the argument in [EMM1] to get more
“generic" counterexamples. First, note that an argument like that in [Mark, Appendix
10] shows that, for any (α, ς) ∈ Q × Q4 ,wehave
# v ∈ Z 4 : Q α
ς (v) = 0 and v ≤T ∼ Cς,αT 2 log T. (15)
Now the same argument as in [EMM1, Lemma 3.15] (see also [Mark, Section 9])
gives: for every ε>0 and any S>0, the set (β,ζ) ∈ R × R 4 for which there exists
T>Ssuch that NQ β ,ζ,(1/4, 1,T) ≥ T 2 (log T ) 1−ε is dense.
Given that S>0, letUS be the set of ordered pairs (γ,ξ) ∈ R × R 4 for which
there exists (β,ζ) ∈ R × R 4 and T>Ssuch that
NQ β ,ζ,(1/4, 1,T) ≥ T 2 (log T ) 1−ε , |β − γ |

INHOMOGENEOUS QUADRATIC FORMS 127
Let the Hamiltonian be the geodesic flow for flat 2-tori. Sarnak [Sar] proved that,
for almost all 2-dimensional flat tori (with respect to Lebesgue measure on the moduli
space of 2-dimensional flat tori), the pair correlation density function converges to the
pair correlation density of a Poisson process. He used averaging arguments to reduce
the pair correlation problem to one of spacing between the values at integers of binary
quadratic forms. This is related to the quantitative Oppenheim problem in the case of
signature (2, 2). One corollary of Theorem 1.3 is that the Berry-Tabor conjecture holds
for the pair correlation of 2-dimensional flat tori under certain explicit Diophantine
conditions.
Similarly, Theorem 1.9 has a corollary in this direction. Let h be a lattice in R 2 .
Also, let α = (α1,α2) ∈ R 2 . Now the eigenvalues of the Laplacian
− =− ∂2 ∂2
−
∂y2 ∂y2 with quasi-periodicity conditions
(17)
φ(x + v) = e 2πi〈α,v〉 φ(x) for all x ∈ R 2 and all v ∈ h (18)
are of the form 4π 2 w + α 2 , where w ∈ h ∗ and where h ∗ is the dual lattice to h. Let
0 ≤ λ0

128 MARGULIS and MOHAMMADI
α ∈ R 2 be given, and define β as above. Suppose that at least one of the following
holds.
(i) The vector β = (β1,β2) is Diophantine.
(ii) There exists N,C > 0 such that, for all triples of integers (p1,p2,q) with
q ≥ 2, we have
max Ai −
i=1,2
pi
>
q
C
q
Then for any interval (a,b) with 0 /∈ (a,b), we have
lim
T →∞ Rh,α(a,b,T ) = c 2
h (b − a). (22)
Hence the spectrum satisfies the Berry-Tabor conjecture for the pair correlation
function.
The case of h = Z 2 was proved by Marklof [Mark]. His approach utilizes results from
the theory of unipotent flows combined with an application of theta sums. We also use
the theory of unipotent flows in our proof; however, our strategy to control the integral
of unbounded functions over certain orbits is dynamical and rests heavily on [EMM1]
and [EMM2].
Outline of the proof. Let Qξ be a quadratic form of signature (p, q),andletn = p+q.
Fix an interval (a,b), andletU⊂Rnbe a “suitably chosen" compact set such that
a

INHOMOGENEOUS QUADRATIC FORMS 129
Quadratic forms. Let n ≥ 3, andletn = p + q, where p ≥ 2. Let {e1,...,en} be
the standard basis for R n . If p ≥ 3, let B be the “standard" form
i
B
i=1
xiei
= 2x1xn +
p
i=2
x 2
i −
n−1
i=p+1
x 2
i . (23)
Let H = SO(B),andlet{at} be the 1-parameter subgroup of H given by ate1 = e −t e1,
atei = ei for 2 ≤ i ≤ n − 1, andaten = e t en. And let K = H ∩ ˆK, where ˆK is the
group of orthogonal matrices with determinant 1. We let dk denote the Haar measure
on K normalized so that K is a probability space.
If (p, q) = (2, 2), welet
B(x1,x2,x3,x4) = x1x4 − x2x3
be the standard form on R4 . This is the determinant on M2(R) if we identify R4 with M2(R). Note that this identification shows that SO(2, 2) is locally isomorphic
to SL2(R) × SL2(R) with the action v → g1vg −1
2 , which leaves the determinant
invariant. We let H = SL2(R) × SL2(R), K = SO(2) × SO(2), andat = (bt,bt),
where bt = diag(e −t/2 ,e t/2 ). We let dk denote the Haar measure on K normalized so
that K is a probability space. We often work with the standard lattice Z 4 and the form
Q, in which case we continue to denote by {at} and K the corresponding 1-parameter
and maximal compact subgroup of SO(Q).
If (p, q) = (2, 1), welet
B(x1,x2,x3) = x1x3 − x 2
2
be the standard form on R3 . This is the determinant on Sym2(R), the space of 2 × 2
symmetric matrices, if we identify R3 with Sym2(R). This identification shows that
SO(2, 1) is locally isomorphic to SL2(R) with the action v → gvtg, where tg is
the transpose matrix. We let H = SL2(R). We let at = diag(e−t/2 ,et/2 ),andwe
let K = SO(2) be the maximal compact subgroup of H. As before, dk denotes the
normalized Haar measure on K.
Let f be a continuous function with compact support on Rn . We define the theta
transform of f by
f ˆ(
+ ξ) =
f (v), (26)
v∈+ξ
where +ξ is any unimodular inhomogeneous lattice in R n . Note that ˆ
f is a function
on the space of inhomogeneous lattices.
(24)
(25)

130 MARGULIS and MOHAMMADI
We fix some more notations. Let n = p + q, andletG = SLn(R) ⋉R n . Let
Ɣ = SLn(Z) ⋉Z n , which is a lattice in G. We have the following, which is similar to
Siegel’s integral formula.
LEMMA 2.1
Let f and fˆ be as above. Let µ be a probability measure on G/ Ɣ which is invariant
under Rn . Then
f ˆ(g)
dµ(g) =
G/ Ɣ
Rn f (x) dx. (27)
Proof
Note that Rn is the unipotent radical of G. Now the lemma follows from Fubini’s
theorem and the fact that µ is Rn-invariant.
We end this section by recalling the definition of the α functions defined on the space
of lattices. Let be a lattice in R n . A subspace L of R n is called -rational if L ∩
is a lattice in L. For a -rational subspace L, letd(L) be the volume of L/(L ∩ ).
For 0 ≤ i ≤ n,define
1
αi() = sup
d(L)
: L is a -rational subspace of dimension i , (28)
and let α() = maxi αi(). Now if ξ = + ξ is an inhomogeneous lattice, let
α(ξ) = α(). There is a constant c = c(f ) depending on f such that, for any
inhomogeneous lattice ξ = + ξ, wehave
ˆ
f (ξ)

INHOMOGENEOUS QUADRATIC FORMS 131
Theorem 1.6 is proved using the following, which is a result of combining Theorems
3.1, A.3, andA.4. We have the following.
THEOREM 3.2 ([EMM1, Theorem 3.5])
Suppose that p ≥ 3 and q ≥ 1. Let f and fˆ be as above. Let ν be any continuous
function on K. Then, for every compact subset D of G/ Ɣ, there exist finitely many
points x1,...,xℓ ∈ G/ Ɣ such that
(i)
(ii)
the orbit Hxi is closed and has finite H -invariant measure for all i;
for any compact set F ⊂ D \
i Hxi, there exists t0 > 0 such that, for all
x ∈ F and t>t0, we have
f ˆ(atkx)
ν(k) dk −
fdµ ˆ
νdk
≤ ε, (31)
K
G/ Ɣ K
where either µ is the G-invariant measure on G/ Ɣ or H ⋉R n x is closed and has
H ⋉R n -invariant probability measure and µ is this measure.
Proof
We may and will assume that φ is nonnegative. Now define
A(r) = ∈ G/ Ɣ : α1() >r . (32)
Let gr be a continuous function on G/ Ɣ such that gr() = 0 if /∈ A(r),gr() = 1
for all ∈ A(r + 1), and0≤ gr() ≤ 1 if r ≤ α1() ≤ r + 1. We have
f ˆ = ( fˆ − fgr) ˆ + fgr. ˆ Note that fˆ − fgr ˆ is a continuous function with compact
support on G/ Ɣ.
Note that H 0 ⋉Rn is a maximal connected subgroup of G. Hence for every
δ>0, there exists r0 such that if H ⋉Rny is a closed orbit of H ⋉Rn in G/ Ɣ with
an H ⋉Rn-invariant probability measure σ ,thenσ (A(r) ∩ H ⋉Rny) r0. Hence for r sufficiently large, we get
fdµ− ˆ ( fˆ − fgr) ˆ
dµ

132 MARGULIS and MOHAMMADI
Hence, if we apply Theorem 3.1, then there is a constant B depending on B1 such that
we get
( fgr)(atkx) ˆ ν(k) dk ≤ B sup |ν(k)| r −β/2
(35)
K
for all x ∈ D.
Now choose r>r0sufficiently large so that B(supk∈K |ν(k)|)r−β/2

INHOMOGENEOUS QUADRATIC FORMS 133
above. Define
f ˜(g
: qξ) =
v∈X(qξ )
f (gv). (36)
The following is an analogue of [EMM2, Theorem 2.3] and will provide us with
the upper bound required for the proof of Theorem 1.9. The proof of this theorem is
the main technical part of this paper and will occupy the rest of this paper.
THEOREM 4.1
Let G, H, K, and {at} be as in Section 2 for the signature (2, 2) case. Let Qξ be a
quadratic form of signature (2, 2) which is Diophantine. Let q ∈ SL4(R) and qξ be
as above. Let ν be a continuous function on K. Then we have
lim sup
t→∞
˜f (atk : qξ)ν(k) dk ≤ f ˆ(g)
dµ(g)
K
G/ Ɣ
K
ν(k) dk, (37)
where µ is the G-invariant probability measure on G/ Ɣ if the homogenous part, Q,
is irrational and where the H ⋉R 4 -invariant probability measure on the closed orbit
is H ⋉R 4 · qξ if Q is a rational form.
Proof of Theorem 1.9
Suppose that Qξ is as in the statement of Theorem 1.9. An argument like that
of [EMM1, Sections 3.4, 3.5] combined with Theorem 4.1 gives: if 0 /∈ (a,b),
then
lim sup
T →∞
NQ,ξ,(a,b,T ) = lim sup ÑQ,ξ,(a,b,T ) ≤ λQ,(b − a)T
T →∞
2 . (38)
This upper bound, combined with the lower bound obtained by Theorem 1.4, proves
Theorem 1.9.
The proof of Theorem 4.1 extensively utilizes results and ideas from [EMM1]
and [EMM2], and we will try to use their terminology and notation for the convenience
of the reader. We recall these theorems and terminology when we need them. Let us
start with the following.
THEOREM 4.2
Let {at} and K be as in Theorem 4.1. Let be any lattice in R 4 . Then for i = 1, 3
and any ε>0, we have
sup αi(atk)
t>0 K
2−ε dk < ∞. (39)

134 MARGULIS and MOHAMMADI
Hence there exists a constant c depending on ε and such that, for all t>0 and
0

INHOMOGENEOUS QUADRATIC FORMS 135
If L is a µ1-quasi-null subspace and if T/2 ≤v L ≤T, then there exists C>0
depending on µ1 such that either π2(v L )

136 MARGULIS and MOHAMMADI
Indeed, an estimate like (43) would actually hold for f ˆ (by virtue of [Sch, Lemma
2]) if A was defined by taking the union over all quasi-null subspaces. But we have
replaced fˆ by f ˜,
and this implies that we can take the union over Q instead. To see
this, consider one of these subspaces, say L1. Assume that ξ ∈ L1 + w1ξ, where
w1ξ ∈ Z4 . Now if there is k ∈ K such that (44) is satisfied with L = L1 and v ∈ Z4 ,
then there is a constant cξ ≥ 1 depending on ξ such that d(atkH) 1/cξδ}. Hence there is c such that (43)
holds.
Now the proof of Theorem 4.1 will be completed if we can show that there is
some η>0 depending on Qξ such that |AL t (δ)|

INHOMOGENEOUS QUADRATIC FORMS 137
Furthermore, if L is a null subspace (of the first type), then
k ∈ K : d(atkL) 0 be small, and let
A L,1
t (δ, ε)
=
k ∈ K : δ1+ε

138 MARGULIS and MOHAMMADI
For simplicity, let ζ = qξ, where q ∈ SL4(R) was chosen such that Q(v) =
B(qv) for all v ∈ R 4 . Fix t > 0 and 0

INHOMOGENEOUS QUADRATIC FORMS 139
(i) there exists λL ∈ such that, if for some λ ∈ there is kφ ∈ EL t (δ, kθ,ε) for
which the plane (L + λ + ζ )φ intersects B(r), then L + λ = L + λL;
(ii) for all λ ∈ , we have maxkφ (at(kθ,kφ)[λ + ζ ]) Lφ ⊥ >c/(δ(1−ε)/2 ).
Furthermore, if (ii) holds, then there exists a computable constant η1 > 0 such that,
for 0 0 such that the function hλ is (C1,β1) good. It follows from the
definition of (C, α)-good functions (see [KM]) that fλ(φ) =(at(kθ,kφ)(v L ∧ (λ +
ζ )) is (C2,β2)-good for some C2,β2 > 0.
Recall now that
c
maxkφ∈E (at(kθ,kφ)[λ + ζ ]) ⊥ Lφ > δ (1−ε)/2 ,
δ1+ε

140 MARGULIS and MOHAMMADI
Note that, for φ ∈ Eλ,wehavefλ(φ)

INHOMOGENEOUS QUADRATIC FORMS 141
let aφ =(at(kθ,kφ)[λ0 + ζ ]) Lφ ⊥. Recall that T/2 ≤vL ≤T, hence aT/2 ≤
w ≤aT. As was mentioned above, K2 acts on the plane spanned by {e123,e124}
by rotation on R 2 , and e123 is the contracting direction of {at}. Also note that we are
assuming that |E L t (λ0)| >δ η2 . Hence, there exist kφ ∈ E L t (λ0) such that
aT δ η2
2 ≤ at(kθ,kφ)[v L ∧ (λ0 + ζ )] = aφat(kθ,kφ)v L . (57)
Note now that we have aφ ≤ r and at(kθ,kφ)v L ≤δ. So we get a ≤ (2rδ 1−η2 )/T,
as we wanted to show.
Before proceeding, let us draw the following corollary. This will be used in the proof
of Theorem 4.1 to control the contribution of “small” subspaces.
COROLLARY 5.5
Let η1 be as in Proposition 5.3, and let η2 1, there
is a δ0 = δ0(M,) such that if δ2, one of the following holds.
(i) The number of quasi-null subspaces of Q of norm between T/2 and T is
O(T 1−τ2 ).

142 MARGULIS and MOHAMMADI
(ii) There exists a split integral form Q ′ with coefficients bounded by a fixed power
of T τ2 and 1 ≤ λ ∈ R satisfying Q − (1/λ)Q ′ ≤T −ρ such that the number
of quasi-null subspaces of Q with norm between T/2 and T which are not null
subspaces of Q ′ is O(T 1−τ2 ).
We refer to [EMM2, Section 10] and also Section 7 of this paper for a more careful
analysis of this theorem. The main ingredient in the proof is the system of inequalities
from [EMM1]. The main difference is that these inequalities are applied to a certain
dilated lattice in 2 R 4 = R 6 .
Let us now outline the rest of the proof. Let Qξ be the inhomogeneous quadratic
form as in the statement of Theorem 1.9. We apply the above theorem with appropriate
parameters ρ,τ2 to be determined later. Let T ≥ 2 be given. Now if (i) above holds
(which is always the case if Q is not EWAS), then we already have a good control on
the number of quasi-null subspaces in question, and we will get the desired control
on the measure of the set
L AL t (δ, ε). Hence, we may assume that (ii) above holds.
Again, if L is not a null subspace of the appropriate approximation of Q, we proceed
as in case (i). So we need to consider the contribution from quasi-null subspaces which
are null subspaces of some rational approximation. In this case, using Lemma 5.3,we
are reduced to the case where only one translate of L has contribution. We then use
the Diophantine property of ξ, and we get a control on the number of such subspaces.
This completes the proof.
We need to fix some more notations before proceeding with the above outline.
If Q is a rational form, we may choose µ1 small enough such that all µ1-quasi-null
subspaces are null subspaces. Also in this case, by replacing Q with a scalar multiple,
we may and will assume that Q is a primitive integral form.
Let T ≥ 2 be a fixed number. Recall from Theorem 5.6 that there are two
possibilities for quasi-null subspaces. The case which requires more study is case
(ii), so let us assume that we are in this case. Let QT = Q ′ (resp., QT = Q) ifQ
is an irrational form (resp., if Q is split integral form), where Q ′ is given as in (ii)
of Theorem 5.6. LetQt(δ, ε, τ2) be the set of all such quasi-null subspaces (resp.,
null subspaces if Q is an rational) which are not exceptional subspaces and such that
A2 δ (•,ε) is nonempty for them. Let L ∈ Qt(δ, ε, τ2) be such a subspace. We further
assume that L is of the first type and that T/2 ≤vL≤T. Since QT is a split integral form, after possibly multiplying by a scalar bounded
by a fixed power of T τ2 , there exists a nonsingular integral matrix such that QT (v) =
B(pv). Furthermore, Theorem 5.6 guarantees that we may choose p such that its
entries are bounded by a fixed power of T τ2 . Recall that the null subspaces of B are
of two types based on the fact that the corresponding vector vL is in V1 or V2. From
now on by a null space of first kind for B, we mean the following.

INHOMOGENEOUS QUADRATIC FORMS 143
Null subspaces of the first type. These are subspaces which are orbits of x11
0
x12 under
0
SL2 × SL2.
Throughout the rest of the section, by a null subspace we mean a rational null
subspace of the first type. Now let M be a rational null subspace of the first type for
B. Such subspaces are characterized as annihilators of primitive row vectors. Hence
an integral basis for M is m 0 0 m
, , where gcd(m, n) = 1. We refer to this basis
n 0 0 n
as the standard integral basis for M.
Recall that L ∈ Qt(δ, ε, τ2) with T/2 ≤vL≤T.Since L is a null subspace of
QT , the subspace M = pL is a null subspace of B. Now let {v1,v2} be the standard
basis for M, and let wi be the unique primitive integral multiple of p−1vi. Then
{w1,w2} is a basis for L. Furthermore, since p is an integral matrix whose entries are
bounded by a fixed power of T τ2 1/2−τ3 , we have T ≤wi ≤T 1/2+τ3 , where τ3 is a
fixed multiple of τ2. We refer to the basis constructed in this way as a τ3-round basis
for L.
Fix 0 δ η2 ,
in particular, that (i) of Proposition 5.3 holds for some λ ∈ . Fix{w1,w2} as a
τ3-round. Note that τ3 is fixed when τ2 is chosen.
Let q −1 λ = v ∈ Z 4 . Now, using Lemma 5.4, there is a constant c = c(q) such
that (v + ξ)L ⊥ < (c(q)δ1−η2 )/T. This and the fact that L is a null subspace of QT
give
|〈wi ,v+ ξ〉QT |≤T 1/2+τ3 · cδ 1−η2
T
cδ1−η2
=
T 1/2−τ3
, (58)
where c is an absolute constant depending on Q. So {〈wi ,ξ〉QT }≤(cδ 1−η2 )/(T 1/2−τ3 ),
where {}denotes the distance to the closest integer. Let us now collect the result of
the above discussion in the following.
LEMMA 5.7
Let ε and τ2 be small, and let L ∈ Qt(δ, ε, τ2). Letkθ ∈ p1(A qL
t (δ, ε)), and suppose
that |E qL
t (δ, kθ,ε,ξ)| >δ η2 with η2 as in Lemma 5.4. In particular, (i) in Proposition
5.3 holds for qL and some λ ∈ . Then {〈wi ,ξ〉QT }≤(cδ 1−η2 )/(T 1/2−τ3 ),
where τ3 is a fixed multiple of τ2 as above. Furthermore, since {w1,w2} is the image of
standard basis {v1,v2} of M = pL, then we have {〈vi , pξ〉B} ≤(cδ 1−η2 )/(T 1/2−τ3 ).
This lemma brings us to the situation where we can now use the Diophantine property
of the vector ξ.
As we mentioned in the brief outline following it, Theorem 5.6 deals with the
Diophantine properties of Q. If Q fails to have the desired Diophantine condition, ξ

144 MARGULIS and MOHAMMADI
needs to be a Diophantine vector thanks to our assumption on Qξ. In what follows,
we make use of this assumption, and we control the number of quasi-null subspaces
for which Lemma 5.7 can hold.
The following is a simple consequence of Definition 1.7 (i.e., the Diophantine
condition). The proof is easy, and we include it for the sake of completeness (and also
for later reference).
LEMMA 5.8
Let x = (x1,...,xn) ∈ R n be a κ-Diophantine vector. Then for any κ ′ > (n+1)κ +1,
we have that, for all β > 0 and all 0

INHOMOGENEOUS QUADRATIC FORMS 145
Proof
We may and will assume that L is of the first type. We continue to use the notations
as in Lemma 5.7. In particular, let M = pL, andlet{v1,v2} be the standard basis
for M. Then Lemma 5.7 implies that {〈vi, pξ〉B} ≤(cδ 1−η2 )/(T 1/2−τ3 ). Recall that
v1 = (m, 0,n,0) and v2 = (0,m,0,n), where gcd(m, n) = 1, and we have
T 1−τ3 ≤v M =m 2 + n 2 ≤ T 1+τ3 , (60)
where τ3 is a fixed multiple of the constant τ2 appearing in Theorem 5.6. We also note
that 〈v1 , pξ〉B = m(pξ)4 − n(pξ)2 and that 〈v2 , pξ〉B(pξ)1 − m(pξ)3.
Let τ ′ 1 ≪ 1/32κ be chosen. Then choose τ2 in Theorem 5.6 such that τ2 T τ ′ 1. Taking τ2 even smaller, we may and will assume that
τ3

146 MARGULIS and MOHAMMADI
Proof
Let η1 be as in Proposition 5.3, andletη2 δ η2 for some kθ ∈ p1(A qL
t (δ, ε)) is O(T 1−τ1 ). On the
other hand, using Theorem 5.6, we see that the number of quasi-null subspaces with
T/2 ≤v L ≤T and A L t (δ, ε) = ∅which are not in Qt(δ, ε, τ2) is O(T 1−τ2 ). The
corollary follows with η = η1/2 and τ = min{τ1,τ2}.
6. Proof of Theorem 4.1
We complete the proof, of Theorem 4.1 in this section. Before proceeding to the proof,
we need the following two statements.
LEMMA 6.1
If Qξ is an irrational (2, 2) form, then the number of 2-dimensional null subspaces,
say L,ofQ such that ξ ∈ v + L for some v ∈ Z 4 is at most four.
Proof
First, note that using [EMM2, Lemma 10.3], we may and will assume that Q is a
rational form. In this case, we actually show that there are at most two such subspaces.
In order to see this, suppose that there are two null rational subspaces of the same
type, say Li, for i = 1, 2 such that ξ ∈ vi + Li, where vi ∈ Z 4 . Now since the Li are
of the same type, they are transversal. Let {w i 1 ,wi 2 } be an integral basis for Li. Then
〈ξ,w i j 〉B ∈ Z for i, j = 1, 2. This (thanks to the transversality of the Li) implies that
ξ is a rational vector, which is a contradiction.
The following is essential to the proof of Theorem 4.1 and proved in Section 7.
PROPOSITION 6.2
The number of quasi-null subspaces with norm between T/2 and T is O(T ).
As we mentioned, this is proved in Section 7. However, for the time being, let us
remark that in the case where Q is a split integral form, this is immediate. Indeed,
in that case we are dealing with null subspaces, and hence we need to show this for
Q = B. As we observed, however, the null subspaces of B are classified by primitive
integral vectors (m, n). Now if L is a null subspace of either type which corresponds
to (m, n), then v L =m 2 + n 2 . Thus the result is obvious.
We now turn to the proof of Theorem 4.1.

INHOMOGENEOUS QUADRATIC FORMS 147
Proof of Theorem 4.1
We may and will assume that ˜f is nonnegative. Now define
A(r) = ∈ X : α1() >r . (63)
Let gr be a continuous function on X such that gr() = 0 if /∈ A(r), gr() = 1
for all ∈ A(r + 1), and0 ≤ gr() ≤ 1 if r ≤ α1() ≤ r + 1. Let the constant c
(depending on f and ξ) beasin(43), and let s ∈ N be a large number. Define
( fgr) ˜ ≤
s = ( fgr)χ ˜
{ : fgr ˜ ()≤c2s } and ( fgr) ˜ >
s = ( fgr)χ ˜
{ : fgr ˜ ()>c2s }. (64)
Now let µ be as in the statement of Theorem 4.1.Sincef˜−fgr ˜ is bounded and
continuous, Theorems A.3 and A.4 imply that
lim sup ( f˜ − fgr)(atkqξ)ν(k) ˜
dk ≤ fdµ ˆ νdk. (65)
t→∞ K
G/ Ɣ K
Hence Theorem 4.1 will be proved if we show that: given ɛ>0, we can choose r0
such that, if r>r0, then lim supt K ˜ fgr dk < ɛ.
Now let ɛ>0 be an arbitrary small number. Let us first control the contribution
of ( fgr) ˜ > s when s is large enough. Let M>0 beanumberfixed(fornow)andlarge
enough such that 1/M < ɛ/6C,where C is a universal constant appearing in (70) and,
in particular, is independent of µ1 in the definition of quasi-null subspaces. Further
assume that M is large enough such that all exceptional subspaces have norm less
than M. Let µ1 in the definition of quasi-null subspace be small enough such that,
for all L ∈ Qt(δ, ε) with vL 0
be large enough such that the conclusions of Theorems 4.2 and 4.5, for this µ1 which
we chose, as well as of Corollaries 5.5 and 5.11, hold for δ = 1/2j whenever j>s.
Recall now that we have
k ∈ K : ˜f (atkξ) >c2 j ⊂ k ∈ K : α13(atkξ) > 2 j
1
∪ Bt
2j
1
∪ At
2j
,
where At(1/2 j ) and Bt(1/2 j ) are as in (43). Let Ct(1/2 j ) ={k ∈ K : α13(atkξ) >
2 j }.Wehave
K
h >
s (atKqξ) dk ≤
s

148 MARGULIS and MOHAMMADI
all j>s,we have
1
Ct
2j
+
Bt
1
2j
+
At
1
2 j
\
L∈Qt ( 1
2j ,ε)
AL
1
t ,ε <
2j 1
. (68)
2 (1+ε/4)j
Let η1 < 1/4 be as in Proposition 5.3, andletη = η1/2. The Conclusions of
Corollaries 5.5 and 5.11 hold with this η and the corresponding τ.Now let Q < t (1/2j ,ε)
(resp., Q ≥ t (1/2 j ,ε)) be the set of quasi-null subspaces in Qt(1/2 j ,ε) with norm less
than M (resp., greater than or equal to M.)Wehave
|
L∈Q < t (1/2j ,ε) AL 1
t (
|
L∈Q ≥ t (1/2 j ,ε) AL t
( 1
2 j ,ε)|≤
i
t i+j e /(2 )
i et /(2i+j+1 )
t i+j e /(2 )
et /(2i+j+1 )
2 j ,ε)| ≤
L |It (δ)| ≤C1 i
L |It (δ)|≤C2
i
1
2 (1+η)j+i/2 +
1
2 (1+η)j+i/2 ,
e −τt
2 (1/2−τ)i+(1−τ)j
where C1 and C2 are absolute constants independent of µ1. The inequality in the
first line above follows from Corollary 5.5 and from the fact that the definition of
Qt(1/2 j ,ε) excludes exceptional subspaces. The inequalities in the second line follow
from Corollary 5.11. Wenowhave
2 j
j>s
L∈Qt
1
2j ,ε
,
(69)
AL
1
1
t ,ε ≤ C
2j 2s 1
+ . (70)
η M
Here C ′ is an absolute constant which depends on C in (70). This inequality together
with (68) gives
K
( fgr) ˜ >
s (atKqξ) dk ≤ C′
+ C
2εs/4
1 1
+
2ηs M
. (71)
Recall that M was chosen such that C/M < ɛ/6. Now we choose s large enough
such that the right-hand side of (71)islessthanɛ/2. The above estimate holds for all
r.
As we mentioned in the proof of Theorem 3.2, There exists r0 = r0(s, ɛ) such
that if r>r0, then µ(A(r)) r0(s, ɛ). We have
lim sup
t→∞
K
( fgr) ˜ ≤
s (atkqξ)ν(k) dk ≤ 2 s lim sup
t→∞
K
gr(atkqξ)ν(k) dk ≤ ɛ/2.
(72)
Thus (72) and(71) give:ifr>r0(s, ε), then
lim sup
t→∞
fgr(atkqξ) ˜ dk < ɛ. (73)
K

INHOMOGENEOUS QUADRATIC FORMS 149
This finishes the proof of Theorem 4.1.
7. Proof of Theorem 1.10
The proof of Theorem 1.10 is very similar to that of Theorem 1.9. Indeed, our study
in this case is simpler as we are dealing with the case where the homogeneous part Q
is rational. Hence, we only need to consider the contribution of null subspaces to the
counting function.
As before, let q ∈ SL3(R) be such that Q(v) = B(qv) for all v ∈ R 3 . Since Q is
rational, we may assume that q is in PGL3(Q). Let p ∈ GL3(Q) be a representative
for q. Let = qZ 3 , and define qξ = q(Z 3 + ξ). As in Section 4, letX(qξ)
be the set of vectors in qξ not contained in qL, where L ⊂ Z 3 is an exceptional
(1-dimensional) subspace for Q. An argument like that in Lemma 6.1 shows that there
are at most three such subspaces when Q is rational and that ξ is an irrational vector.
For any continuous compactly supported function f on R 3 , define
f ˜(g
: qξ) =
v∈X(qξ )
f (gv). (74)
As in previous discussions, we reduce the proof of Theorem 1.10 to the following
theorem.
THEOREM 7.1
Let G, H, K, and {at} be as in Section 2 for the signature (2, 1) case. Let Qξ be a
quadratic form of signature (2, 1) as in the statement of Theorem 1.10.Letq∈ SL3(R)
and qξ be as above. Let ν be a continuous function on K. Then we have
lim sup
t→∞
K
˜f (atk : qξ)ν(k) dk ≤
G/ Ɣ
f ˆ(g)
dµ(g)
K
ν(k) dk, (75)
where µ is the H ⋉R 3 -invariant probability measure on the closed orbit H ⋉R 3 ·qξ.
The proof of this theorem is very similar to that of Theorem 4.1. We will use the
same notations as those in the previous sections for the sake of simplicity. The main
notational difference to bear in mind is that in previous sections L denotes a 2dimensional
subspace, where in this section L is a 1-dimensional (null) subspace.
As before, we need to study the subsets of K where the function ˜f is “large”.
We start by recalling the following. There is c>0 such that, for all large t and small

150 MARGULIS and MOHAMMADI
0 c
⊂ k ∈ K : α1(atkqξ)
δ
> 1
∪ k ∈ K : α2(atkqξ) >
δ
1
. (76)
δ
We fix t and δ as above. Let us make two important remarks before we continue.
Remarks 7.2
(i) Using reduction theory of the orthogonal group, we see that if α2(atkqξ) >
1/δ, then actually α1(atkqξ) > 1/δ. Hence we only need to study the contribution
coming from α1.
(ii) The fact that Q is a rational form implies that there exists δ0 depending only on
Q such that if 0

INHOMOGENEOUS QUADRATIC FORMS 151
For any null subspace L of Q, the subspace pL is a (rational) null subspace of
B. We let vL denote the standard basis for pL. For any such L, let AL t (δ) be the
corresponding set in (77). We have the following.
LEMMA 7.3
There exists 0

152 MARGULIS and MOHAMMADI
Some important properties of these intervals were proved in [EMM2] and recalled in
Lemma 5.1. What is important for us in Section 7 is property (ii) in Lemma 5.1.This
property, tailored to our current assumptions, gives
|A L
t
−t
L e δ
1/2 (δ)| ≤|It (δ)| ≈ . (81)
T
Proof of Theorem 7.1
Let M be a large number which is fixed for now. Let t>0 be a large number. Assume
also that j is a large number. We assume throughout that η = η1, where η1 is as in
Lemma 7.3. LetL0,L1 be nonexceptional null subspaces such that {〈v Lk , pξ〉B} <
1/2 j for k = 0, 1 and such that v L0 ≤v L1 are minimal with these properties.
Indeed, we fix any two such subspaces if there are more than two subspaces satisfying
these conditions. Assume that e t /2 j+i0+1 ≤v L 2 jη2 , where η2 is as in Lemma 7.3. Now using this
lemma, we see that, for all i ≤ i0, the number of null subspaces L with e t /2 j+i+1 ≤
v L i0, the subspaces of norm at most e t /2 i+j have
no contribution to the set At(1/2 j ), that is, A L t (1/2j ) =∅for all such subspaces L.
Hence if we use this fact and (81), we have: if j is such that Case 1 holds, then
we have
Lnull
AL
1
t
2j
≤
i
≤ 1
2 j(1+η)
et /(2i+j+1 )≤vL

INHOMOGENEOUS QUADRATIC FORMS 153
Now, if we argue just as in Case 1, we get
L null
AL
1
t
2j
≤
i
et /(2i+j+1 )≤vL

154 MARGULIS and MOHAMMADI
of G, and let K be the maximal compact subgroup of H . The question which is
addressed in this section is that of the equidistribution of sets of the form atKx,
where x ∈ G/ Ɣ and where A ={as : s ∈ R} is a suitable 1-parameter subgroup
of H. This is a well-studied question (see [Sh2]). The main tools in the analysis are
indeed Ratner’s theorem on classification of unipotent flow-invariant measures on
G/ Ɣ, and linearization techniques for the action of unipotent groups on G/ Ɣ which
were developed by Dani and Margulis [DM].
Let H and W be closed subgroups of G. Following Dani and Margulis [DM],
define X(H,W) ={g ∈ G : Wg ⊂ gH}. We recall the following.
THEOREM A.1 ([DM, Theorem 3])
Let G be a connected Lie group, and let Ɣ be a lattice in G. Let U ={ut} be
an Ad-unipotent 1-parameter subgroup of G. Let φ be a bounded continuous function
on G/ Ɣ. Let D be a compact subset of G/ Ɣ, and let ε>0 be given. Then
there exist finitely many proper closed subgroups H1 = H1(φ,D,ε),...,Hk =
Hk(φ,D,ε) such that Hi ∩ Ɣ is a lattice in Hi for all i, and compact subsets
C1 = C1(φ,D,ε),...,Ck = Ck(φ,D,ε) of X(H1,U),...,X(Hk,U), respectively,
for which the following holds. For any compact subset F of D \
i CiƔ/Ɣ
there exists T0 > 0 such that, for all x ∈ F and T>T0, we have
1
T
φ(utx) dt − φdg
0.
Note that the standard representation of H 0 on Rn is irreducible, hence H 0 is a
maximal connected subgroup of H ⋉Rn . Note also that as H 0 is a maximal connected
subgroup of SLn(R), we have that H 0 ⋉Rn is a maximal connected subgroup of G.

INHOMOGENEOUS QUADRATIC FORMS 155
Now let Ɣ be a lattice in G. Then Ɣ ∩ R n is a lattice in R n . We let = ϑ(Ɣ) =
Ɣ/ Ɣ ∩ R n (this is a lattice in SLn(R)). We abuse the notation and let ϑ also denote
the projection from G/ Ɣ onto SLn(R)/. We have the following.
LEMMA A.2
Let x ∈ G/ Ɣ. Then the orbit Hϑ(x) is closed in SLn(R)/ if and only if H ⋉R n x
is closed in G/ Ɣ.
Proof
Note that closed orbits of H ⋉R n (resp. H ) have a finite H ⋉R n -invariant (resp.,
H -invariant) measure by [Mar86, Section 3]. Let x = (g, v)Ɣ. Note that H ⋉R n x =
ϑ −1 (Hϑ(x)), hence if the Hϑ(x) is closed, then so is H ⋉R n x. Suppose now that
H ⋉R n x is closed. Then Ɣ1 = H ⋉R n ∩ (g, v)Ɣ(g, v) −1 is a lattice in H ⋉R n ,
and hence Ɣ1 ∩ R n is a lattice in R n and Ɣ1/Ɣ1 ∩ R n is a lattice in H. Hence Hϑ(x)
is closed, as we wanted.
The following is special case of [EMM1, Theorem 4.4].
THEOREM A.3
Let the notation be as above. Further assume that is a lattice in H 0 ⋉R n . Let φ
be a compactly supported continuous function on H 0 ⋉R n /. Then for every ε>0
and any bounded measurable function ν on K, and for every compact subset D of
H 0 ⋉R n /, there exist finitely many points x1,...,xℓ ∈ H 0 ⋉R n / such that
(i) the orbit H 0 xi is closed and has finite H 0 -invariant measure for all i;
(ii) for any compact set F ⊂ D \
i Hxi, there exists s0 > 0 such that, for all
x ∈ F and s>s0, we have
φ(askx) ν(k) dk −
K
H 0 ⋉R n /
φdg
K
νdk
≤ ε. (89)
We also need a slight variant of [EMM1, Theorem 4.4]. This is the content of the
following.
THEOREM A.4
Let G, H, K, and Ɣ be as above. Let A ={as : s ∈ R} also be as above. Let φ be
a compactly supported continuous function on G/ Ɣ. Then for every ε>0 and any
bounded measurable function ν on K, and for every compact subset D of G/ Ɣ,there
exists finitely many points x1,...,xℓ ∈ G/ Ɣ such that
(i) the orbit H ⋉R n xi is closed and has finite H ⋉R n -invariant measure for all
i;

156 MARGULIS and MOHAMMADI
(ii) for any compact set F ⊂ D \
i H ⋉Rn xi, there exists s0 > 0 such that, for
all x ∈ F and s>s0, we have
φ(askx) ν(k) dk −
K
G/ Ɣ
φdg
K
νdk
≤ ε. (90)
Proof
The proof of this theorem goes along the same lines as in [EMM1, Section 4]. Let U be
as defined above. Let Hi = Hi(φ,KD,ε) and Ci = Ci(φ,KD,ε) for i = 1,...,k
be given as in Theorem A.1 corresponding to U.For1 ≤ i ≤ k, define
Gi = g ∈ G : Kg ⊂ X(Hi,U) . (91)
Note that the group generated by
k∈K k−1Uk is H 0 , as U is not contained in any
proper normal subgroup of H and K is the maximal compact subgroup of H. Now let
g ∈ Gi. Then k−1Uk ⊂ gHig−1 for all k ∈ K. Hence, H 0 ⊂ gH 0
i g−1 . Now as H 0
acts irreducibly on Rn , the only possibilities for gH 0
i g−1 are H 0 and H 0 ⋉Rn . Thus
we have
gH 0
i g−1 ⊂ H 0 ⋉R n , for any g ∈ Gi, 1 ≤ i ≤ k. (92)
We also note that if g ∈ G is such that gH 0 g −1 ⊂ H 0 ⋉R n , then g ∈ NSLn(R)(H 0 ) ⋉
R n . This fact and (92) say that if g1,g2 ∈ Gi for some i, then g −1
1 g2 ∈ NSLn(R)(H 0 ) ⋉
R n . Since H 0 ⋉R n is of finite index in NSLn(R)(H ) ⋉R n , we get that Gi can be
covered by finitely many cosets of H 0 ⋉R n . Hence there are finitely many points
x1,...,xℓ ∈ G/ Ɣ such that the H ⋉R n xi are closed and have a finite H ⋉R n -
invariant measure for all 1 ≤ i ≤ ℓ and such that
GiƔ/ Ɣ ⊂
H ⋉Rn xi. (93)
i
1≤i≤ℓ
Note that since the X(Hi,U) are analytic submanifolds of G and since K is connected,
we have that, for any g ∈ G \
i GiƔ/ Ɣ,
k ∈ K : kg ∈
X(Hi,U)Ɣ = 0. (94)
i
Now (93), (94), and the fact that Ci ⊂ X(Hi,U), give
|{k ∈ K : kx ∈
CiƔ/ Ɣ}| = 0 (95)
i

INHOMOGENEOUS QUADRATIC FORMS 157
for any x ∈ F ⊂ D \
i H ⋉Rnxi. Now if we apply [EMM1, Lemma 4.2], then
there exists an open subset W ⊂ G/ Ɣ such that
i CiƔ/ Ɣ ⊂ W and such that
|{k ∈ K : kx ∈ W }| 0 such that
1
T
φ(uty) − φdg
0 be any sufficiently small number, and let T > 2. There exists a rational
3-dimensional subspace U of 2 R 4 with a reduced integral basis of norm at most
T τ whose projections into V2 have norm less than T τ−1 such that one of the following
holds.
(a) The restriction of Q (6) to U is anisotropic over Q, in which case (i) in Theorem
5.6 holds.
(b) The restriction of Q (6) to U splits over Q, in which case (ii) in Theorem 5.6
holds. Furthermore, in this case Q ′ (as in loc. cit.) is proportional to f (U,U ⊥ ),
where U ⊥ is the orthogonal complement of U with respect to Q (6) and where f
is as in Lemma 4.3. Moreover, the number of quasi-null subspaces with norm
between T/2 and T which are not in U or U ⊥ is O(T 1−τ ).
Let us denote by Q (3) (v) the restriction of Q (6) to U. As we mentioned before, this is
a form with signature (2, 1) and U is a rational subspace. If L is a quasi-null subspace
with T/2 ≤v L ≤T, then we have Q (6) (v L ) = 0. Hence if L is a quasi-null
subspace in U, then Q (3) (v L ) = 0. In other words, Proposition 6.2 follows from the
following.

158 MARGULIS and MOHAMMADI
PROPOSITION B.2
Let Q0(x,y,z) = 2xz − y 2 be the standard quadratic form of signature (2, 1) on
R 3 . Let = gZ 3 ,whereg ∈ GL3(Q). Let T > 0 be a large parameter, and let τ
be a sufficiently small parameter. Assume that the entries of g are rational numbers
whose nominator and denominators are bounded by a fixed power of T τ , and also
suppose that |det g| ≤T τ . Further assume that the length of the shortest vector in
is c = O(1). Then if T is large enough, we have
# w ∈ P () : Q0(w) = 0 and w ≤T 0. Note that C is dilation-invariant. Let λ = disc() 1/3 , and define
1 = 1/λ. The lattice 1 is unimodular, and the length of the shortest vector in
1 is at least c/λ. Let¯g ∈ PGL3(Q) denote the image of g. Indeed, 1 = ¯gZ 3 . The
counting problem in (98) follows if we show that
# w ∈ P (1) : Q0(w) = 0 and w ≤ T

INHOMOGENEOUS QUADRATIC FORMS 159
Hence we see that C ∩P (1) ⊂ Ɣmv0. As is a finite set, we only need to consider
Ɣmv0.
Let e1 = (1, 0, 0), and let N0 ⊂ H be the stabilizer of e1. Let BR ={h ∈ H :
h · e1 ∈ CR}, and let BR = BR/N0. Now let χR denote the characteristic function of
CR. Define the following:
FR(h) =
χR(hγ v), for h ∈ H. (100)
Ɣm/Ɣm∩N
This is a function on H/Ɣ.Now using the well-developed machinery for counting
problems using the mixing property (see in particular [BO] and[MS]), there exists
ε>0 depending only on the spectral gap of H and τ such that
FR(e) = |N0/N0 ∩ Ɣ|
vol(BR)
|H/Ɣ|
1 + O(vol(BR) −ε ) , (101)
where vol denotes the H -invariant Haar measure on H/N. Hence we have
#(Ɣmv0 ∩ CR) ≤ FR/v0(e) = O(R/v0). (102)
Since v0 ≥c/λ, this finishes the proof.
Acknowledgments. We would like to thank J. Marklof for reading the first draft and
for many helpful comments. We also thank the anonymous referees for their remarks
and suggestions. The second author would like to thank A. Eskin for conversations
regarding these remarks.
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Margulis
Department of Mathematics, Yale University, New Haven, Connecticut 06520, USA;
margulis@math.yale.edu
Mohammadi
Department of Mathematics, University of Chicago, Chicago, Illinois 60637, USA;
amirmo@math.uchicago.edu