Applications of Symplectic Geometry to Hamiltonian Mechanics

Applications of Symplectic
Geometry to Hamiltonian
Mechanics
Samuel Thomas Lisi
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Mathematics
New York University
January 2006
Helmut H. W. Hofer

Dedicated to my grandparents, Thomas and Helvi Lisi.
iii

Acknowledgements
First and foremost, I thank my advisor Helmut Hofer. His great enthusiasm for
symplectic geometry couldn’t help but be contagious. His insight and intuition
continue to be an inspiration to me. I also thank him for sharing many insights
into the non-mathematical aspects of being a mathematician.
I thank Casim Abbas for much support and guidance. Many of the ideas in
Chapter 5 trace their roots to conversations with Professor Abbas. In addition,
his feedback has been most helpful, and his advice on expository writing has
been invaluable.
I thank Joe Coffey for many helpful discussions, and for his encouragement.
I am especially grateful to him for explaining dividing sets to me, and for helping
me to understand Legendrian surgery.
I am grateful to Richard Hind for clarifying for me the implications of Stein
domains.
I would like to acknowledge the help, guidance and support of Chris Wendl,
Richard Siefring, Breno Madero, Joel Fish and Barney Bramham. I have learned
much of what I know from our Courant Symplectic group, and for this I am
very grateful.
I thank all of my friends and family and everyone else who has made this
possible. I thank my parents and brother for their constant encouragement and
support. Without them, none of this would have been possible. Finally, and
most deeply, I am grateful to Leah Tiisler for always being there and supporting
me through everything.
iv

Abstract
In this thesis, we consider three applications of pseudoholomorphic curves to
problems in Hamiltonian dynamics. In a first part, we prove an existence result
for homoclinic orbits on a contact-type, critical energy level of an autonomous
Hamiltonian, provided that the level is Hamiltonian displaceable. To do this, we
transform the problem into a problem of Lagrangian intersection Floer theory.
This involves a construction due to Mohnke [34] and some ideas from Legendrian
surgery. In particular, we prove a generalization of Séré’s result [36] on the
existence of homoclinic orbits for an autonomous Hamiltonian system.
In a second part, we develop a theory of pseudoholomorphic curves into a
singular contact manifold, which represents the critical level of an autonomous
Hamiltonian. We show that a pseudoholomorphic half-plane is asymptotic to
a homoclinic orbit. Furthermore, in a non-degenerate case, this convergence
is of an exponential nature. This result is a first step towards understanding
the change in contact homology under Legendrian surgery. Such a surgery
formula would enable the computation of contact homology for every contact
three manifold.
Finally, we lay the groundwork for an energy quantization result for pseudoholomorphic
planes with a weaker energy bound than in the existing theory.
This is a question related to the problem of understanding the compactification
of the space of generalized pseudoholomorphic curves as envisioned by Hofer
and studied by Abbas, Cieliebak and Hofer in [3]. This is part of an ongoing
project with Casim Abbas and Helmut Hofer.
v

Contents
Dedication iii
Acknowledgements iv
Abstract v
List of Figures viii
1 Introduction and Main Results 1
1.1 Geometric constructions and Hamiltonian dynamics . . . . . . . 3
1.1.1 Lagrangian intersections and Homoclinic orbits for Hamiltonian
systems . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Lagrangian intersections and the Weinstein conjecture . 4
1.2 PDE techniques and Hamiltonian dynamics . . . . . . . . . . . 4
1.2.1 Homoclinic orbits by PDE methods . . . . . . . . . . . . 4
1.2.2 Finding periodic orbits by means of generalized pseudoholomorphic
curves . . . . . . . . . . . . . . . . . . . . . 6
2 Preliminaries 9
2.1 Contact and symplectic manifolds . . . . . . . . . . . . . . . . . 10
2.1.1 Contact manifolds . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Legendrian submanifolds . . . . . . . . . . . . . . . . . . 12
2.1.3 Weinstein domains . . . . . . . . . . . . . . . . . . . . . 12
2.2 Legendrian surgery . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 The surgery presentation . . . . . . . . . . . . . . . . . . 14
2.2.2 Weinstein’s handle attaching construction . . . . . . . . 19
2.2.3 Attaching a critical handle performs −1 surgery . . . . . 21
3 An existence result for homoclinic orbits 24
3.1 The main theorem and overview . . . . . . . . . . . . . . . . . . 24
3.2 Proof of our result . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Applications to the Weinstein conjecture . . . . . . . . . . . . . 34
vi

4 Pseudoholomorphic curves in the singular level 38
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 A compactification of the singular level . . . . . . . . . . . . . . 39
4.2.1 Construction of the compactification . . . . . . . . . . . 39
4.2.2 A local model of the singularity . . . . . . . . . . . . . . 41
4.2.3 A local model for the neighbourhood of a homoclinic orbit 46
4.3 Type (I) and (II) ˜ J-holomorphic curves . . . . . . . . . . . . . . 49
4.3.1 The singular almost complex structure . . . . . . . . . . 49
4.3.2 Type (I) pseudoholomorphic curves . . . . . . . . . . . . 50
4.3.3 Type (II) pseudoholomorphic curves . . . . . . . . . . . 50
4.4 Asymptotic behaviour of a Type (II) finite energy half-plane . . 52
4.4.1 Convergence to a homoclinic orbit . . . . . . . . . . . . . 52
4.4.2 Exponential rate of convergence . . . . . . . . . . . . . . 63
4.5 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Energy quantization for pseudoholomorphic curves with a relaxed
area bound 70
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 “Renormalization” by the Reeb flow . . . . . . . . . . . . . . . . 73
5.3 Proof of the result . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3.1 The boundary value problem . . . . . . . . . . . . . . . . 75
5.3.2 Compactness of disks . . . . . . . . . . . . . . . . . . . . 76
5.3.3 Fredholm theory . . . . . . . . . . . . . . . . . . . . . . 84
5.4 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Bibliography 91
vii

List of Figures
1.1 Relationship between homoclinic orbits, periodic orbits and Reeb
chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Weinstein’s local handle model . . . . . . . . . . . . . . . . . . . 19
2.2 The global handle attaching construction . . . . . . . . . . . . . 20
3.1 Perturbing the critical level F = 0. . . . . . . . . . . . . . . . . 32
3.2 The Lagrangian L . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1 Local model near the singular Legendrian . . . . . . . . . . . . . 46
viii

Chapter 1
Introduction and Main Results
In this thesis, we apply pseudoholomorphic curve methods to the study of Hamiltonian
dynamics. We focus on two main problems : the study of homoclinic
orbits in a Hamiltonian system, and the Weinstein conjecture. Our pseudoholomorphic
curve methods fall into two broad categories. In a first part of this
thesis, we exploit a geometric construction due to Mohnke [34]. This allows us
to transform dynamical questions into a question of whether a given immersed
Lagrangian is embedded or not. In a second part, we use more analytical methods
to study pseudoholomorphic curves. In each part, we address both the
question of homoclinic orbits and the Weinstein conjecture.
The first type of orbit we consider are periodic orbits. The Weinstein conjecture
claims that for any Reeb vector field on any closed contact manifold,
there exists a periodic orbit. In dimension 3, much progress has been made,
though the conjecture is, at present, open. In higher dimensions, the body of
work is quite sparse. In this work, we prove the Weinstein conjecture holds
for a certain class of contact manifolds realized as hypersurfaces in symplectic
manifolds, for any dimension. We also prove the Weinstein conjecture holds for
a contact form on a three-manifold if there exists a pseudoholomorphic plane
with a weaker energy bound than previously required.
The second type of orbits we consider are homoclinic orbits (homoclinic to
a rest point). We recall that in an autonomous Hamiltonian system, given by
a Hamiltonian function H, critical points of H are rest points for the Hamiltonian
flow. An orbit which is asymptotic to a rest point, both in forward and
in backwards time, is called a homoclinic orbit. The existence of a transverse
homoclinic orbit is quite interesting from a dynamical perspective, as it then
implies the existence of rich dynamics nearby (by Smale’s horseshoe construction).
Pseudoholomorphic curves are an essential tool in modern Symplectic Geometry.
Introduced by Gromov in 1985 [20], they allow one to see symplectic
1

geometry as a generalization of Kähler geometry. A pseudoholomorphic curve
is a solution to a certain partial differential equation, which can be seen as
a non-linear analogue to the Cauchy–Riemann equations of complex analysis.
Pseudoholomorphic curve theory combines methods from many different areas
of mathematics : non-linear functional analysis, elliptic partial differential
equation theory on manifolds, topology, complex geometry, classical symplectic
geometry. The study of these curves provides the essential ingredient of many
results in symplectic and contact geometry, including results on Lagrangian
intersections, symplectic embeddings and Hamiltonian dynamics.
A first example of such an application to Hamiltonian dynamics comes from
Floer’s proof of a conjecture of Arnol ′ d. This gives a lower bound on the number
of periodic orbits for a Hamiltonian system, in terms of the topology of the
underlying manifold. To obtain this result, Floer defined a homology theory,
similar in flavour to Morse theory, in which pseudoholomorphic curves serve as
the analogue to gradient flow lines [15].
In 1993, Hofer [21] introduced the appropriate notion of pseudoholomorphic
curve into a contact manifold (which is a geometric object that corresponds,
roughly speaking, to a compact energy level in a classical Hamiltonian system).
He established that a non-compact pseudoholomorphic curve with a certain
energy bound is asymptotic to a periodic orbit of the associated Hamiltonian
system. Building on this, Eliashberg, Givental and Hofer introduced symplectic
field theory and its simpler cousin, contact homology, which are invariants of
symplectic cobordisms between contact manifolds [13]. These Floer homologylike
invariants are differential graded algebras, generated by periodic orbits with
a differential that counts pseudoholomorphic curves connecting them.
In this thesis, we build on these results in several directions. In Chapter 2, we
introduce the notation we use for the remainder of this work, and we also provide
some background constructions that will be useful in the remaining sections. In
Chapter 3, we present results in Hamiltonian dynamics obtained by means of
a geometric construction. This construction allows us to reduce the problem
to one of showing a Lagrangian cannot be embedded. The non-embeddedness
of this Lagrangian will then come from an argument using pseudoholomorphic
disks with boundary in the Lagrangian. In Chapter 4, we introduce a class of a
contact manifolds with singular contact forms that correspond to critical levels
in an autonomous Hamiltonian system. We then develop a singular pseudoholomorphic
curve theory to study homoclinic orbits on the corresponding critical
level in the Hamiltonian system. In Chapter 5, we study pseudoholomorphic
planes into symplectizations of contact manifolds, but with a weaker energy
condition than has been considered in the literature (e.g. [21, 6]). We prove
that such curves have an energy threshold. This is closely related to Abbas,
Cieliebak and Hofer’s programme to solve the Weinstein conjecture in dimen-
2

sion 3 by means of studying a generalization of the class of holomorphic curves.
The remainder of this chapter summarizes the main results and discusses the
connection to existing work.
1.1 Geometric constructions and Hamiltonian
dynamics
1.1.1 Lagrangian intersections and Homoclinic orbits for
Hamiltonian systems
Our main result in Chapter 3 is a new existence result for homoclinic orbits : a
singular energy surface of contact type with precisely one hyperbolic singularity
and finite displacement energy contains a homoclinic orbit. Séré’s important
result [36] concerning such hypersurfaces in Euclidean space is then a corollary
of this more general result. The proof uses a powerful construction from Mohnke
[34] and some ideas from Legendrian surgery.
The main result is as follows :
Theorem 3.1.2. Let (W0, ω0) be a symplectic manifold (with boundary) of dimension
2n, and let F be a Morse function on W0 so that F has a unique critical
point x0 of index n. Suppose there is a Liouville vector field Y , i.e., LY ω = ω,
gradient-like for F . Suppose furthermore that the Hamiltonian vector field of
F , XF , has a hyperbolic zero at x0.
We normalize F by the conditions F (x0) = 0, and dF [Y ] ≤ 1.
Suppose that there exists S > 0 and a symplectic manifold (W, ω), symplectically
aspherical and of bounded geometry, so that (F −1 [−S, 0), ω0) symplectically
embeds in (W, ω) and furthermore, the embedding, restricted to F −1 (−S/2), induces
an injection from the fundamental group of F −1 (−S/2) to the fundamental
group of W .
Suppose furthermore that there exists a Hamiltonian diffeomorphism Φ of W ,
with compact support and Hofer norm ||Φ||, displacing the image of F −1 [−S, 0).
Then, our original Hamiltonian vector field XF has an orbit homoclinic to x0
of action bounded above by ||Φ||/(1 − e −S/2 ).
We obtain Séré’s result [36] as a corollary to this. Also as a corollary, we
obtain that any compact, contact-type singular hypersurface with precisely one
hyperbolic singularity in a subcritical Stein domain admits a homoclinic orbit.
We note that our methods are very different from Séré’s. The proof in
[36] uses a variational approach. Our approach here relates the problem to
3

one of symplectic topology. We exploit the fact that if we have an embedded
Lagrangian submanifold, and we can displace it from itself by a Hamiltonian
diffeomorphism, then there must be the bubbling off of either a disk or a sphere
in the moduli spaces of pseudoholomorphic strips, as in Lagrangian intersection
Floer homology.
1.1.2 Lagrangian intersections and the Weinstein conjecture
In Chapter 3, we also introduce a construction that allows us to prove the
Weinstein conjecture for Hamiltonian displaceable contact-type hypersurfaces of
symplectic manifolds. This again builds on Mohnke’s Lagrangian construction.
We obtain,
Theorem 3.3.1. Let (M, ξ) be a contact manifold with contact form λ.
Suppose that there exists a symplectic manifold (W, ω) of bounded geometry and
symplectically aspherical, and S > 0 so that we may symplectically embed ι :
([−S, S] × M, d(e s λ)) ↩→ (W, ω) and furthermore, this embedding induces an
injection on the fundamental groups.
If there exists a Hamiltonian diffeomorphism Φ of W with compact support and
finite Hofer norm ||Φ|| displacing the image of [−S, S] × M, then there exists a
periodic orbit of action bounded above by 4||Φ||/(e S − e −S ).
In particular, this result applies for all subcritically Stein fillable contact
manifolds. This also overlaps with a result of Liu and Tian [31].
1.2 PDE techniques and Hamiltonian dynamics
In Chapters 4 and 5, we turn to PDE methods in order to study pseudoholomorphic
curves, with an eye to applications in Hamiltonian dynamics.
1.2.1 Homoclinic orbits by PDE methods
In Chapter 4, we study pseudoholomorphic half-planes with image in the symplectization
of a singular contact manifold, of dimension 3. This singular contact
manifold is a model of the critical level of an autonomous Hamiltonian system.
Such Hamiltonians arise also in a handle surgery construction, due to Weinstein
(which we discuss in Chapter 2). The use of pseudoholomorphic half–planes to
understand homoclinic orbits (in Euclidean space) was first considered by Hofer
and Wysocki [22].
4

In a first part, we introduce a construction to compactify the critical level
(with the critical point deleted) by gluing in a Legendrian knot, L. This then
provides a model for the critical level as a closed contact manifold, M. The
construction also gives us a contact form, singular along the Legendrian knot.
We then establish a local model near this singular Legendrian.
From this local model, we are led to consider a special class of singular
almost complex structures on R × M. These are singular along R × L. The
local model, however, allows us to describe the almost complex structure in a
neighbourhood of R × L.
Finally, we introduce two types of pseudoholomorphic curve. The first type,
Type (I) pseudoholomorphic curves, are pseudoholomorphic maps into R × M
in the least restrictive sense : they are pseudoholomorphic everywhere u(z) /∈ L.
In other words, their derivative is complex linear everywhere the image carries
an almost complex structure. The second type, Type (II) pseudoholomorphic
curves, is much more restrictive : we impose a condition to make the curve
compatible with the local model, in a sense we make precise in Chapter 4.
Nevertheless, it is an elliptic problem, and satisfies some useful properties. In
particular, we show that in certain special circumstances, the two classes are
actually the same.
We prove that a Type (II) pseudoholomorphic half-plane is asymptotic to
a homoclinic orbit. Furthermore, we show that with slightly stronger boundary
conditions, such a half-plane converges at an exponential rate. A very nice
relationship between homoclinic orbits and Legendrian chords allows us to circumvent
many difficulties in this setting.
We may state the main result as follows, where H = {s + it ∈ C | s ≥ 0} is
the right half-plane :
Theorem 4.4.4. Suppose that ũ : H → R × M is a Type (II) ˜ J holomorphic
curve, has finite energy and satisfies the boundary condition at infinity :
dist(u(0 + it), L) → 0 as |t| −→ ∞.
Let T = u ∗ dλ. Then, for any sequence R ′ k → ∞ there is a subsequence Rk so
that
u(Rke it ) → x(T t/π)
where x(t) is a homoclinic orbit of the (singular) Reeb vector field. The convergence
is in C ∞ ([−π/2, π/2], M).
We also obtain a result, more technical to state, that if the asymptotic
limit is a transverse homoclinic orbit, then there exist coordinates in which the
rate of convergence is exponential. We show that this rate of approach may
be described in terms of an eigenvalue (and corresponding eigenvector) of a
5

Figure 1.1: This figure shows a homoclinic orbit on the critical level, together
with the corresponding periodic orbit and Legendrian chord on nearby levels.
This diagram is merely a simplistic sketch, to illustrate the type of phenomena
we are interested in.
self-adjoint, unbounded operator. These results are given precisely in Theorem
4.4.12.
A motivation for the study of these half-planes, in addition to the dynamical
interest of finding homoclinic orbits, is that the change in contact homology
under Legendrian surgery is related to the occurrence of homoclinic orbits on
an associated critical energy level (see Figure 1.1). This current work is part of
a larger goal to encode these changes in terms of a contact homology for this
critical level, which would include the homoclinic orbits among the generators.
Ultimately, this might provide a means of computing the change in contact
homology across Legendrian surgery. This would be of particular interest in
dimension three, for which there is a Legendrian surgery presentation of every
contact manifold ([10]). This approach might then lead to a surgery formula for
contact homology.
Another interesting point is that the Floer complex of such a theory would
carry extra structure. Transverse homoclinic orbits admit a concatenation operation.
Corresponding to this, we should also have a gluing of half-planes. This
represents an extra operation on the Floer complex, of a nature not studied
before.
1.2.2 Finding periodic orbits by means of generalized
pseudoholomorphic curves
In Chapter 5, we consider the behaviour of infinite energy pseudoholomorphic
planes. We recall that in all current applications of pseudoholomorphic curves
to contact geometry, we restrict our attention to finite energy curves. These are
curves for which
E(ũ) := sup{ ũ ∗ d(φλ) | φ : R → [0, 1] with φ ′ ≥ 0} < ∞.
6

This energy was introduced in [21], and is the assumption necessary for the
existing compactness theory, as in [6].
In this work, we consider the case of E(ũ) = ∞, but impose instead :
0 < u ∗ dλ < ∞.
We refer to this weaker energy as the contact area since it measures the symplectic
area of the projection of T u to the contact structure. We note that this
condition means that the R–factor of the curve is no longer a proper map. Such
infinite energy curves then behave very wildly, as compared to the finite energy
curves.
In this work, we prove that such curves have an energy threshold :
Theorem 5.1.1. Suppose ũ : C → R × M is pseudoholomorphic with respect
to an almost complex structure ˜ J, adjusted to the contact form λ, and has finite
contact area :
0 < u ∗ dλ = C < ∞.
C
We also impose the technical condition that πλ T u = 0.
Then, there exists a periodic orbit of the Reeb vector field with action 0 < T ≤ C.
The key ingredient of the proof is a construction of renormalization. This
construction makes precise the observation, due to Hofer, that such infinite
energy curves are merely finite energy curves that have been “smeared” out by
the Reeb flow.
The technical condition that πλ T u never vanish is merely introduced to
simplify the analysis. We note that πλ T u also satisfies a Cauchy-Riemann type
equation, and so, if not identically zero, its zeros are isolated and have positive
multiplicity. In practice, then, this condition may be checked by algebraic
means.
The more general case, in which πλ T u may have zeros, introduces a greater
number of technical problems. We include, in Chapter 5, a discussion of the
ingredients of the proof in the more general case. We also remark that preliminary
work indicates that C should be equal to a period of the Reeb vector
field.
This result is part of an ongoing research project with Casim Abbas and
Helmut Hofer. Our eventual goal is to understand precisely in what sense we
may speak of these curves as being asymptotic to a periodic orbit.
A description of such planes is necessary to understand the compactification
of Hofer’s generalized pseudoholomorphic curve theory. These generalized pseudoholomorphic
curves satisfy a PDE which resembles the non-linear Cauchy-
Riemann equations, but with an extra term involving a harmonic form. A
7

compactification of the space of such objects is needed for the Abbas-Cieliebak-
Hofer program to prove the Weinstein conjecture in dimension three (begun in
[3]).
8

Chapter 2
Preliminaries
We are interested in the situation in which we have a 2n dimensional Weinstein
domain (W, ω, Y, F ). That is, we have a symplectic manifold (W, ω) and a
smooth Hamiltonian function F : W → R whose level sets are contact–type.
Then, a level set of F for a regular value is a contact manifold with a preferred
choice of contact form, iY ω| F −1 (c). The associated Reeb vector field is a positive
multiple of the Hamiltonian vector field, and so questions about qualitative
behaviour of the dynamics associated to the Hamiltonian vector field (existence
of periodic orbits, ...) can be answered by studying the dynamics of the Reeb
vector field.
The problem we are interested in is to understand what happens at a critical
level for F . This is no longer a submanifold of W . Nevertheless, assuming
only finitely many critical points, after removing the critical points of F , we
obtain a non–compact contact manifold. In this situation, a new qualitative
behaviour can emerge : a homoclinic orbit. A homoclinic orbit is an orbit of
the Hamiltonian vector field that converges, both in forward and backward time,
to the same critical point of F .
This problem is of interest both in the study of Hamiltonian dynamics, and
also in order to understand what happens to contact homology after performing
Legendrian surgery.
We will begin by reviewing some results and constructions in contact geometry.
This will allow us to define our terminology and fix sign conventions.
A deep observation that traces back to Eliashberg [12], is that contact manifolds,
unlike their symplectic cousins, allow for “cut-and-paste” constructions.
Much work in contact geometry has been accomplished by these methods. This
thesis lays the foundations for a project to understand how these cut-and-paste
constructions interact with pseudoholomorphic curves.
In this chapter, we will discuss Legendrian surgery through two different
constructions. We will also prove some neighbourhood theorems that will be
9

useful in our later analysis.
2.1 Contact and symplectic manifolds
2.1.1 Contact manifolds
A contact manifold (M 2n−1 , ξ) is a smooth, odd-dimensional manifold, M, together
with a hyperplane bundle ξ ⊂ T M, called the contact structure. This
hyperplane bundle satisfies a non-integrability condition : locally, we can find a
one form λ so ξ = ker λ and λ ∧ (dλ) n−1 is a volume form. We say the contact
structure is co-orientable if it can be realized globally as the kernel of a one-form
λ with λ ∧ (dλ) n−1 a volume form on M.
Any one-form λ so that ker λ is a contact structure (equivalently, λ∧dλ > 0)
is called a contact form. We note that if λ is a contact form generating the
contact structure ξ, then fλ is also, for any f : M → (0, ∞). The contact
structure, ξ, is a symplectic vector bundle over M, with symplectic structure
given by dλ. We note that even though λ is not canonical, the conformal class
of dλ|ξ is.
A diffeomorphism Ψ : M1 → M2 between two contact manifolds, (M1, ξ1)
and (M2, ξ2) is called a contact diffeomorphism, or a contactomorphism, if
Ψ∗ξ1 = ξ2. If ξi = ker λi, then this is equivalent to requiring the existence
of a function f : M1 → (0, ∞) so that Ψ ∗ λ2 = fλ1.
On a contact manifold (M, ξ), given the choice of a contact form, we introduce
the Reeb vector field, X. This vector field satisfies dλ(X, ·) = 0 and
is normalized so λ(X) = 1. An essential point is that the Reeb vector field
depends on the choice of contact form. Once we have a choice of contact form,
and thus of Reeb vector field, we have a splitting of T M = RX ⊕ ξ. We let
πλ denote the projection from T M → ξ corresponding to this splitting. Where
there is no ambiguity, we will sometimes omit the subscript, and write π.
In this work, we will be interested in understanding the dynamics of the
Reeb vector field. There are two main motivations for this. The first is that
Reeb dynamics are closely related to Hamiltonian dynamics (for instance, the
work in [27] answers questions about geodesic flow on S 2 ). The second is that a
better understanding of all possible Reeb dynamics associated to a given contact
structure will shed light on contact geometry.
Relation to symplectic geometry
Contact manifolds arise naturally as boundaries of symplectic manifolds. Indeed,
we say that a contact manifold M is a contact–type hypersurface in a
10

symplectic manifold (W, ω) if there exists a vector field, Y , defined in a neighbourhood
of M, so that LY ω = ω and Y is transverse to M. This induces a
contact structure on M by taking the maximal symplectic sub-bundle of T M.
We call any vector field Y with the property that LY ω = ω a Liouville vector
field, a conformally symplectic vector field or an expanding vector field. We note
that all three terms are used in the literature, but occasionally with different
sign conventions (e.g. Weinstein, in [37], uses the opposite sign convention).
If M is a contact-type hypersurface in W , and Y is a Liouville vector field
transverse to M, we obtain a preferred choice of the contact form by λ = iY ω|M.
We note that by Cartan’s formula (and since ω is closed), d iY ω = LY ω = ω,
so this form is the restriction of a primitive of ω to M. We note that any two
Liouville vector fields transverse to a given contact-type hypersurface M induce
isotopic contact forms. By Gray’s theorem (which we discuss below), the two
contact structures are contactomorphic.
Given such a contact–type hypersurface, M, there are many contact–type
hypersurfaces M ′ nearby. Let Ψt be the flow of the vector field Y and f : M →
R, with small enough C 0 norm that Ψf(x) is defined for all x ∈ M. Then we may
define another submanifold of W by M ′ = {Ψf(x)(x) | x ∈ M}. This manifold is
diffeomorphic to M, but the induced contact form, pulled back to M is given
by e f(x) λ.
An important result on the stability of contact structures is as follows :
Theorem 2.1.1 (Gray’s Theorem). Let (M, ξ0) be a closed contact manifold.
Suppose that ξt are contact structures on M for t ∈ [0, 1], varying smoothly in
t. Then, there exists a one parameter family of diffeomorphisms Ψt : M → M,
t ∈ [0, 1] so that Ψ0 = Id and Ψt∗ξt = ξ0.
Equivalently, if ξt = ker λt, then Ψt ∗ λt = ftλ0, where ft : M → (0, ∞) is a
smoothly varying family of functions with f0 ≡ 1.
Every contact manifold (M, ξ) with contact form λ may be realized as a
contact-type hypersurface in some (non-compact) symplectic manifold. Indeed,
we take W = R × M with the symplectic form ω = d(e a λ), where a is the
coordinate on the R component. M embeds as {0}×M with transverse Liouville
vector field Y = ∂
∂a .
Contact manifolds, much like symplectic manifolds, admit no local invariants.
Indeed, we have a number of neighbourhood theorems. The first is a
Darboux chart theorem:
Theorem 2.1.2. let M and M ′ be contact manifolds with contact forms λ and
λ ′ respectively. For any p ∈ M and p ′ ∈ M ′ , there exist neighbourhoods U and
U ′ and a diffeomorphism ψ : U → U ′ so that ψ(p) = p ′ and ψ ∗ λ ′ = λ.
11

2.1.2 Legendrian submanifolds
In symplectic topology, an interesting class of submanifolds are the Lagrangian
submanifolds. Likewise, in contact geometry, we have an analogous concept.
Suppose (M, ξ) is a contact manifold of dimension 2n−1. We say a submanifold
L ⊂ M is Legendrian if L is n − 1 dimensional and T L ⊂ ξ. Equivalently, T L
is a Lagrangian sub-bundle of the symplectic vector bundle (ξ, dλ). (Observe
that even though this statement makes use of the contact form λ, it is actually
independent of λ since it only relies on the conformal class of dλ|ξ.)
In this work, we will focus primarily on closed contact 3-manifolds. In this
case, a (closed) Legendrian submanifold will be a link.
An important property of Legendrian submanifolds is that they admit a
neighbourhood theorem.
Theorem 2.1.3 (Legendrian Neighbourhood Theorem). Let (M 2n−1 , λ) be a
contact manifold with preferred choice of contact form. Let ℓ be a Legendrian
submanifold of M. Then, for a small neighbourhood U of the Legendrian ℓ and
for ɛ > 0, sufficiently small, we have a diffeomorphism Ψ : U → (−ɛ, ɛ) × T ∗ ℓ,
with Ψ ∗ λ = dt + Θ, where t is the coordinate in the R variable and Θ is the
canonical one form on T ∗ ℓ.
We recall that the canonical one form Θ can be written as Θ = pi dqi, where
qi are coordinates on the manifold and pi are coordinates on the fibre.
This is [4, Thm. 2.2.4]. A slightly weaker form, which only shows that a
Legendrian neighbourhood is contactomorphic to a standard model, is given by
[16, Cor. 2.28].
The Legendrian neighbourhood theorem is the cornerstone of many “cutand-paste”
constructions on contact manifolds. We will discuss how this allows
us to construct new contact manifolds by attaching handles, creating connected
sums and, in the three dimensional case, performing Dehn surgery on the manifold,
and creating a (canonical) contact structure on the new manifold.
2.1.3 Weinstein domains
In order to simplify our bookkeeping in dealing with the cut-and-paste constructions,
we need the notion of a Weinstein domain.
A Weinstein domain consists of a a symplectic manifold (W 2n , ω), a Liouville
vector field Y and a Morse function F , with Y gradient–like for F . In other
words, we require LY ω = ω, that the critical points of F are precisely the zeros
of Y and furthermore, that the zeros of Y are non-degenerate. We also require
that Y be transverse to the boundary of W . By using Y , we introduce the
12

positive and negative boundary of W . We require then that the positive and
negative boundaries of W be regular level sets of F .
We observe that we follow the opposite sign convention from Weinstein, who
first introduced the concept in [37], under the name of “Liouville pair”. We will
denote a Weinstein domain by the tuple (W, ω, Y, F ).
Let x0 be a critical point for F . Let W u be the unstable manifold for the
flow of Y , and W s the stable manifold. Then, W u and W s are co-isotropic
and isotropic, respectively. It follows from our definition that all of the critical
points of F have Morse index at most n. We also observe that every regular
level set of F is a contact–type hypersurface in W .
If the negative boundary of W is empty, we call the Weinstein domain a
Weinstein filling. A special case of a Weinstein domain is a Stein domain —
indeed, in that case we may take Y = ∇F and ω = − d( d(F ◦ J)), where F is
the plurisubharmonic function.
A contact manifold M is said to be Weinstein fillable if there exists a Weinstein
filling for which M is the positive boundary. We remark that we differ
here from the terminology used in [8], where Cieliebak uses the terms Weinstein
domain and relative Weinstein domain where we use, respectively, Weinstein
filling and Weinstein domain.
2.2 Legendrian surgery
We will now present Legendrian surgery in two different ways. The first description
makes it clear that Legendrian surgery is Dehn surgery along a knot
together with a contact-theoretic construction to construct a contact structure
on the surgered manifold, unique up to contactomorphism. The uniqueness
(and a short-cut to existence) of the construction will require some background
on convex surfaces.
The second construction, due to Weinstein, will realize this same surgery by
means of attaching a handle to our contact manifold. This will have the useful
side-effect of constructing a Weinstein domain providing an oriented symplectic
cobordism from our old contact manifold to the new contact manifold, or viceversa,
depending on the sign of the surgery. We remark that this construction is
more general than our application makes explicit — it allows for the construction
of a connected sum of two contact manifolds, for instance.
We will finally show that Legendrian ±1 surgery may be realized by means
of the handle attaching construction due to Weinstein [37]. This formulation of
Legendrian surgery will be the most useful to us, and is of interest in its own
right.
13

2.2.1 The surgery presentation
Given a framed knot in a three dimensional manifold and a rational number r
(possibly infinite), we may construct a new manifold obtained by performing
surgery on that knot. The remarkable fact is that given a Legendrian knot in a
contact 3–manifold M, and if 0 = r = 1/q for some 0 = q ∈ Z, then the manifold
M ′ obtained by performing 1/q surgery on M also carries a contact structure
(which is independent of the choices made in the construction). Furthermore,
away from a neighbourhood of the Legendrian knot, the two manifolds are
contactomorphic.
We will first discuss the construction involved in 1/q surgery, and refer to
the key results from the literature that establish some of the key properties of
this Legendrian surgery. We will sketch a proof of the result that 1/q surgery
can be realized by a sequence of +1 or −1 surgeries. In our exposition, we will
follow the notation of Ding, Geiges and Stipsicz, and will cite some of their key
results on Legendrian surgery presentations of contact 3–manifolds.
We refer the interested reader to the papers by Ding, Geiges and then also
Stipsicz : [9, 10, 11]. In these papers, Ding and Geiges introduce p/q surgery
(rather than just 1/q as we discuss here). They also show in [10] that every
contact three manifold may be obtained by 1/q surgery on a link in the standard
tight S 3 . Then, in [11], an algorithm is given to turn these r = 1/q surgeries
to a sequence of +1 and −1 surgeries. The authors of these papers, and others,
have used these constructions to find various contact structures that are tight,
but with interesting properties with regards to symplectic fillability. Many of
the ideas and constructions are due to Eliashberg and then to Gompf [12, 19].
In order to perform 1/q surgery, we must start with a framed knot. We note
that in a contact 3-manifold (M, ξ = ker λ), a Legendrian knot L comes with
a canonical framing from the contact structure. Indeed, a framing is given by
the choice of a vector field along K, positively transverse to ξ. Any two choices
will then be homotopic.
To simplify the exposition of the construction (at least conceptually), we
will introduce some results of Giroux and Honda on convex surfaces. Furthermore,
these results will be key in establishing that that the resultant (M ′ , ξ ′ ) is
independent of the choices made. Honda has written a good survey article that
provides a self-contained collection of definitions and results [29].
We will first introduce the concept of a characteristic foliation. Suppose
Σ ↩→ M is an embedded surface in a contact 3–manifold (M, ξ). Then, we
introduce a singular foliation F on Σ by taking, at each point p ∈ Σ where ξ is
not tangent to Σ, the line in TpΣ that lies in ξp. The singularities are the points
where ξp = TpΣ. In other words, F(p) = ξp ∩ TpΣ.
The characteristic foliation is important because it describes the neighbour-
14

hood of Σ up to contactomorphism [17]:
Proposition 2.2.1. Let ξ and ξ ′ be two contact structures that induce the same
characteristic foliation on Σ. Then ξ and ξ ′ are isotopic through contact structures,
where the isotopy fixes Σ.
A (closed) convex surface in a contact 3-manifold (M, ξ) is an embedded
(closed) surface Σ ↩→ M for which there exists a transversal contact vector field
Z. We say that a vector field Z is contact if the flow of Z preserves the contact
structure ξ. Equivalently, if ξ = ker λ, we have LZλ = fλ for some function
f : M → R. We will only consider the case of closed surfaces. We note that
among embedded closed surfaces, the property of being convex is C ∞ -generic.
Given a convex surface Σ, we introduce the dividing set,
ΓΣ := {x ∈ Σ | Z(x) ∈ ξx}.
This will be a collection of properly embedded smooth curves in Σ. The isotopy
class of ΓΣ does not depend on choice of Z.
We say that a singular foliation F is adapted to a dividing set ΓΣ if there
exists a contact structure ζ defined in a neighbourhood of Σ so that ΓΣ is
the dividing set for ζ and F is the characteristic foliation for ζ. (We remark
that Giroux has characterized which singular foliations come as characteristic
foliations [17]. Furthermore, we may intrinsically characterize which pairs of
characteristic foliation and dividing set are adapted [10]).
The main interest of convex surfaces and their dividing sets comes from a
result of Giroux’s [17] :
Theorem 2.2.2 (Giroux’s Flexibility Theorem). Assume Σ is convex with characteristic
foliation F0, transverse contact vector field Z and dividing set ΓΣ. Let
F1 be another singular foliation adapted to ΓΣ. Then, there exists an isotopy φt
of Σ in M, t ∈ [0, 1] so that :
1. The isotopy fixes the dividing set : φt|ΓΣ = Id for all t.
2. The isotopy is transverse to the transverse contact vector field : φt(Σ) ⋔ Z
for all t ∈ [0, 1].
3. The final surface φ1(Σ) has characteristic foliation F.
The essential point of this theorem is that the dividing set contains the
complete information about the contact structure in a neighbourhood of the
convex surface.
For our application, we need the following result, the key ideas of which are
due to Kanda [30] and to Makar-Limanov [32], but whose formulation in these
terms is due to Honda [28]. This is Proposition 4.1 in [29] (and a proof is also
provided there) :
15

Theorem 2.2.3. Let n ∈ Z. Suppose we have a solid torus S 1 × D 2 with
boundary T 2 = ∂(S 1 × D 2 ), a candidate dividing set Γ T 2 and a characteristic
foliation on T 2 that is adapted to Γ T 2. Suppose furthermore that we have :
(a) the number of components of the dividing set #Γ T 2 = 2
(b) with respect to a framing of the torus in which the meridian has zero slope
and the longitude (x, y) = (0, 0) has infinite slope, slope(Γ T 2) = 1/n.
Then there exists a tight contact structure on S 1 × D 2 , unique up to isotopy
fixing the boundary, realizing T 2 as a convex surface, and so that its dividing
set is given by our candidate Γ T 2.
We note that by Giroux’s flexibility theorem, the requirement of the existence
of an adapted characteristic foliation is a compatibility condition on the
dividing set : what it is does not matter since it is isotopic to any other one.
We also observe that this result is only stated for n > 0 in [29]. Nevertheless,
the result is true for all n ∈ Z. Indeed, suppose we have a contact structure ξ
on S 1 × D 2 so that the slope of Γ T 2 is 1/n. Then, we push ξ forward by means
of a diffeomorphism of the solid torus corresponding to m Dehn twists along the
meridian to obtain a contact structure ξ ′ on S 1 × D 2 . We compute then that
the dividing set for this contact structure has slope 1/(n + m).
Remark 2.2.4. A standard model for each of these is given by the following
contact forms on S 1 × D 2 :
and by
αn = cos nφ dx + sin nφ dy for n = 0
α0 = αstd = dx + y dφ.
We recall that, by the Legendrian neighbourhood theorem, a Legendrian
knot has a tubular neighbourhood admitting a standard local model. We will
take the model :
Nδ = {(x, y, φ) ∈ R × R × S 1 | x 2 + y 2 ≤ δ 2 }
α0 = dx + y dφ
L0 := {(0, 0, φ) | φ ∈ S 1 }.
Here, and in the following, we will take S 1 = R/Z. Then, by the Legendrian
neighbourhood theorem, there exists a δ0 > 0 and a diffeomorphism onto its
image Ψ : N2δ0 −→ M so that Ψ ∗ λ = α0 and Ψ(L0) = L. We will let ξ0 = ker α0
be the contact structure on Nδ.
16

We will now work in our local model. We consider the boundary of Nδ,
which we denote Tδ := ∂Nδ. This is now a convex torus. Indeed, we observe
that the vector field Z := x ∂ ∂ + y is a contact vector field transverse to Tδ.
∂x ∂y
A meridian m of this torus is given by φ = const. We will take as longitude l
one of the two curves that form the dividing set, l = {(0, 1, φ)}. A 1/q surgery
takes the meridian m and maps it to m+ql. The exact behaviour of l under the
mapping does not matter, since any two choices differ by a Dehn twist, which
then extends to a diffeomorphism of the solid torus.
In order to construct a contact structure on the surgered manifold, we will
need to study a tubular neighbourhood of Tδ. We let A := N2δ0 \ int Nδ0, a
tubular neighbourhood of the convex surface Tδ. We now construct a map
g : A → A so that Tδ0 is mapped to itself, as is T2δ0. We want the map g to
send m to m + ql. This will be our gluing map. We now consider the contact
form on A given by α1 = g∗α0. We let ζ := ker α1 be the corresponding contact
structure. We must now extend ζ to the solid torus N2δ0.
The goal now is to extend ζ to a smooth contact structure on the solid torus
N2δ0. To show that this is possible, we will use the result on convex surfaces
we cited earlier. By our construction, we have a characteristic foliation and
dividing set on the boundary of N2δ0. It now follows by Theorem 2.2.3 that
there exists a unique contact structure on N2δ0. We note that the existence
of such a filling may be shown by more direct, computational methods — the
use of Theorem 2.2.3 will become much more useful in establishing uniqueness.
(The computations, in a special case, will be carried out below.)
Our surgered manifold will then be given by
M ′ = M \ Ψ(Nδ0) ∪ N2δ0/ ∼
where we identify the points w ∈ N2δ0 \ Nδ0 with the points Ψ(g(w)) ∈ M.
Since Ψ : N2δ0 → M and g : N2δ0 \ Nδ0 → N2δ0 \ Nδ0 are contactomorphisms,
the contact structure on M \ Ψ(Nδ0) ⊔ N2δ0 given by ξ ⊔ ζ descends to a smooth
contact structure ξ ′ on M ′ .
An important property of this construction is that up to contactomorphism,
the resultant (M ′ , ξ ′ ) only depends on (M, ξ) and on L. In other words, the
result is independent of the choices of Ψ, of g and of continuation of ζ. This is
a result of Ding and Geiges [9, Proposition 7]. The proof of this result hinges
on the observation that up to contactomorphism, we may assume that the two
surgeries are performed in the same Legendrian model neighbourhood. The
result then follows as an application of Theorem 2.2.3.
Ding and Geiges also establish that 1/q and −1/q Legendrian surgery are
inverses of each other. Specifically, if we perform 1/q Legendrian surgery on M
at L to obtain M ′ , L ′ , then, performing −1/q surgery on M ′ at L ′ gives us M
and L again, up to contactomorphism [9, Proposition 8].
17

To prove this result, by the uniqueness of the surgery, we need only consider
the behaviour of the surgery in the local model. We then show that the boundary
of local model surgered twice has the same dividing set as the the boundary of
the local model with no surgery. The result now follows by Theorem 2.2.3.
The same idea can be pushed further to obtain the following result [9, Proposition
9]:
Proposition 2.2.5. If (M ′ , ξ ′ ) is obtained from (M, ξ) by 1/q surgery for q = 0,
we may also obtain (M ′ , ξ ′ ) by a sequence of |q| applications of q/|q| surgeries.
This proposition is essential in that it allows us to focus all of our attention
on ±1 Legendrian surgery. This surgery, however, can be realized by a handleattaching
construction due to Weinstein [37].
±1 surgery, in coordinates
We may carry out ±1 surgery directly, in coordinates, without appeal to the
theory of dividing sets, if we do not concern ourselves with uniqueness.
Indeed, we consider a Legendrian neighbourhood D 2 × S 1 with contact form
given by :
cos φ dx + sin φ dy.
We delete the Legendrian {(0, 0)} × S 1 and introduce exponential polar coordinates
by x = e s cos θ and y = e s sin θ. We then obtain a deleted neighbourhood
diffeomorphic to R − × S 1 × S 1 with contact form
α− = e s (cos(φ − θ) ds + sin(φ − θ) dθ) .
We will now compactify R − × S 1 × S 1 with this contact structure ξ = ker α−
to S 1 × D 2 , by gluing in a different Legendrian.
Indeed, we introduce ψ = φ − θ and η = θ. Let now
We then have
u = 2e s/2 cos ψ v = 3e s/2 sin ψ.
α+ := du + v dη = e s/2 α−.
The new contact form α+ extends smoothly over the u = v = 0 circle.
We now observe that this construction maps a transverse translate of the
Legendrian {θ = φ} (parallel to the dividing set) to {ψ = 0, η = φ} and the
meridian {φ = 0} to {η = −ψ}. On R − × S 1 × S 1 , we have orientations given
by ds ∧ dφ ∧ dθ = ds ∧ dψ ∧ dη. Thus, we have performed −1 surgery.
We have now performed −1 surgery explicitly from α− to α+. To perform
+1 surgery, it suffices to reverse these steps.
18

N +
δ
L +
N −
δ
Figure 2.1: The local handle attaching model. The arrows represent the Liouville
vector field, Y . The shaded region corresponds to Wδ (see Figure 2.2).
L −
L −
N −
δ
L +
2.2.2 Weinstein’s handle attaching construction
Weinstein’s general construction takes a 2n−1 dimensional contact manifold M
and attaches a k ≤ n handle to it, obtaining a manifold M ′ . The construction
realizes the new manifold as a contact manifold, and furthermore provides a
symplectic cobordism between the old manifold and the new one. More specifically,
the construction realizes a Weinstein domain (W, ω, Y, F ) so that M is the
negative boundary of W and M ′ is the positive boundary. Furthermore, in the
case of an n handle, which we call a critical handle, there are Legendrian spheres
L and L ′ in M and M ′ so that M \ nbd(L) is contactomorphic to M ′ \ nbd(L ′ ).
Let (M, ξ) be a 2n − 1 dimensional contact manifold and let L be a Legendrian
sphere in M. Let λ be a contact form with ξ = ker λ. By the Legendrian
neighbourhood theorem, there is a neighbourhood of L which is contactomorphic
to a standard model. Our construction will take place entirely within this
neighbourhood. This is illustrated in Figure 2.1.
As in the discussion of surgery, we let Nδ be the model neighbourhood of a
Legendrian knot. We then have a contactomorphism Ψ : N2δ → U, where U is
19
N +
δ

M −
Figure 2.2: The global attaching construction. The shaded region is given by
the local model in Figure 2.1. The unshaded part is given by I × (M \ Nδ).
a neighbourhood of the Legendrian knot. We let U ′ be the image of Nδ.
The idea of the construction is very straightforward, though managing all of
the details becomes cumbersome. We first split our contact manifolds M into
two pieces : M = (M \U ′ )∪N2δ/ ∼, where ∼ is the identification provided by Ψ.
We will now construct our symplectic manifold (W, ω) in two pieces. The first
piece will be a trivial symplectic cobordism, W1 := [a, b]×(M \U ′ ), ω = d(etλ). The Liouville vector field here will be ∂ . This is organized in Figure 2.2.
∂t
The second piece, (W2, ω2), will be obtained by means of a local construction
in R2n , which we will discuss next. The straightforward, but technical, part of
the construction is then to arrange a smooth identification of points near the
boundaries of W1 and of W2. For the full details, we refer the reader to [37].
We now present the local model in R2n . We consider R2n with the standard
symplectic form ω0 = dxi ∧ dyi. Let H : R2n → R by (x, y) ↦→ |x| 2 − 1/2|y| 2 .
Then, Y := ∇H = 2xi ∂ ∂ − yi is a Liouville vector field. We observe
∂xi ∂yi
that H has an index n critical point at 0 and no other critical points. Thus,
H = c = 0 is a contact–type submanifold of R2n . We have then that the induced
contact form is given by:
λ = 2xidyi + yidxi
restricted to the tangent space to H = c. We also observe that the intersection
of any regular level set H = c = 0 with the stable or unstable manifold of 0
for the flow of Y is a Legendrian sphere in the contact manifold H = c. Let us
denote this Legendrian sphere by L c .
20
M +

We let Ψt be the flow of the vector field Y . We observe that if c, d > 0
or c, d < 0, then Ψ induces a contactomorphism from the level H = c to the
level H = d. Furthermore, if c > 0 and d < 0, we have that Ψ induces a
contactomorphism from H −1 (d) \ L d to H −1 (c) \ L c .
Let L − = {(x, y) | x = 0, |y| 2 = 2} be the intersection of the stable manifold
of 0 for the vector field Y with the level set H = −1. This is a Legendrian sphere
in H = −1. We will now consider a tubular neighbourhood of this Legendrian
N −
δ := {(x, y) | |x| 2 − 1/2|y| 2 = −1 and |x| ≤ δ}. Let L + = {(x, 0) | |x| 2 = 1}
be the intersection of the unstable manifold of Y with H = 1.
Then, there exists a neighbourhood N +
δ of L+ so that Ψ induces a diffeomor-
phism from N −
δ \ L− → N +
δ \ L+ . We note that N +
δ = {(x, y) | |x|2 − 1/2|y| 2 =
1 and |y| ≤ δ ′ }, where δ ′ = δ ′ (δ) is a monotone function of δ, with δ ′ (0) = 0.
We now let Wδ be the region in R2n between N − +
δ and N δ and bounded by the
flow lines of Ψ through the boundary of N −
δ . This now gives us our local model
(Wδ, ω0, Y, H) that we glue in to the symplectization of M \ U.
We refer the reader to [37] and to [8] for details in the gluing. Weinstein
and Cieliebak, in these articles, also show that the result of the construction is
independent of the choices made (up to symplectomorphism and contactomorphism).
2.2.3 Attaching a critical handle performs −1 surgery
We will now show how this handle attaching construction realizes −1 Legendrian
surgery. To show this, we need only consider the local model. We also note that
we may choose a slightly different model neighbourhood to replace the N −
δ from
above. Indeed, we will take instead :
N −
δ := {(x, y) ∈ R 2n | |x| ≤ δ and |y| = 1}.
For δ small enough, this is transverse to the vector field
Y = ∂ ∂
2xi − yi .
∂xi ∂yi
We then have that N −
δ is a contact submanifold of R 2n , with the contact form
We also take :
λ − = 2xi dyi + yi dxi.
N +
δ := {(x, y) ∈ R 2n | |x| = 1 and |y| ≤ δ}.
We have that L − := {(0, y) | |y| = 1} and L + := {(x, 0) | |x| = 1}.
21

We observe that N −
δ \ L− is diffeomorphic to N + √ \ L
δ + . We also have that
N −
δ and N + √ are contactomorphic to the model neighbourhoods we used in the
δ
previous section (hence our re-use of the same names).
From these coordinates, we have a framing of the solid torus N −
δ
that Z := x1 ∂
∂x1
. We have
+ x2 ∂
∂x2 is a contact vector field transverse to ∂N − . With
respect to this framing, we have then that the dividing set has slope 1.
On N + , we have a different framing. In these coordinates, the dividing set
has slope −1. This then amounts to a −1 surgery along the knot L − . We can
also see this since, after a performing a Dehn twist on each local model, we
obtain the same local model as in our previous example, in which we worked
out −1 surgery explicitly in coordinates.
Computations
It will be convenient for our later work to study the behaviour of the surgery in
explicit coordinates. We have a diffeomorphism Ψ from R − ×S 1 ×S 1 → N − \L −
by (s, θ, φ) ↦→ (e s cos(θ), e s sin(θ), cos(φ), sin(φ)). This pulls back the contact
form λ − = 2xi dyi + yi dxi to the form
α − = e s (cos(θ − φ) ds + 2 sin(θ − φ) dφ − sin(θ − φ) dθ).
Remark 2.2.6. We observe this isn’t quite one of the standard forms on a Legendrian
neighbourhood. However, it is merely a Dehn twist away from being
of the form dx + y dφ. Indeed, this follows if we take x = e s cos(θ − φ) and
y = e s sin(θ − φ).
Now, the flow of Y induces a diffeomorphism from the image of Ψ to N + \L + .
Thus, we obtain a diffeomorphism from R − × S 1 × S 1 toN + \ L + by
(s, θ, φ) ↦→ (cos(θ), sin(θ), e s/2 cos(φ), e s/2 sin(φ)).
This pulls λ + back to the contact form
α + = e s/2 (cos(θ − φ) ds + 2 sin(θ − φ) dφ − sin(θ − φ) dθ).
We also observe that the flow of Y induces a diffeomorphism from the image
of Ψ to the critical level Msing := {(x, y) | |x| 2 − 1
2 |y|2 = 0}. The induced contact
form is
αsing = ce 2s/3 (cos(θ − φ) ds + 2 sin(θ − φ) dφ − sin(θ − φ) dθ).
22

At this point, we observe a fact that will be key in the following section. If
we take the map
x = 3
2 e2s/3 cos(θ − φ)
y = 1
2 e2s/3 sin(θ − φ)
ψ = θ + φ
Then, we obtain that αsing is the pull back of dx+y dψ. As our map is a double
covering, this shows that this singular level has a neighbourhood that double
covers the standard Legendrian neighbourhood.
This will turn out to be important for our work in Chapter 4. We will
exploit this double cover to be able to study pseudoholomorphic curves into the
symplectization of the singular level in this Weinstein domain.
23

Chapter 3
An existence result for
homoclinic orbits
3.1 The main theorem and overview
In this section, we prove the existence of a homoclinic orbit on a compact,
contact–type critical level of a Hamiltonian, assuming that a neighbourhood of
this level is displaceable by a Hamiltonian with finite oscillation. Furthermore,
we provide an upper bound for the length of this homoclinic orbit. This is then
a generalization of a result of Séré’s, which gives this result in R2n (where the
condition on displaceability is a consequence of compactness).
First, we recall some important definitions.
We recall that a symplectic manifold (W, ω) is said to be symplectically
aspherical if ω|π2(M) = 0. This condition is useful in that it guarantees that all
pseudoholomorphic spheres are constant.
We also recall that a symplectic diffeomorphism Φ : W → W is said to
be a Hamiltonian diffeomorphism if it is the time 1 map of a time dependent
Hamiltonian. To such a Φ, we may associate the Hofer norm, denoted ||Φ||.
This is given by taking the infimum of the oscillation over all Hamiltonians that
generate Φ :
||Φ|| = inf
Ht generates Φ
1
0
(sup Ht − inf Ht) dt
W W
Finally, we recall the definition of bounded geometry, essentially due to
Gromov [20] :
Definition 3.1.1 (Bounded geometry). We say that the non-compact symplectic
manifold (W, ω) is geometrically bounded if there exists an almost complex
structure J such that ω(·, J·) is a Riemannian metric on W whose sectional
24

curvature is bounded above and whose injectivity radius is bounded below.
We also refer to such almost complex structures as having bounded geometry.
The result we will prove is :
Theorem 3.1.2. Let (W0, ω0, Y, F ) be a Weinstein domain of dimension 2n,
so that F has a unique critical point x0 of index n and so that the corresponding
Hamiltonian vector field XF has a hyperbolic zero at x0. Let us take F to be
normalized by the conditions F (x0) = 0, and dF [Y ] ≤ 1
Suppose that there exists S > 0 and a symplectic manifold (W, ω) of bounded
geometry and symplectically aspherical so that (F −1 [−S, 0), ω0) symplectically
embeds in (W, ω) and furthermore, the embedding, restricted to F −1 (−S/2), induces
an injection from the fundamental group of F −1 (−S/2) to the fundamental
group of W .
Suppose furthermore that there exists a Hamiltonian diffeomorphism Φ of W ,
with compact support and finite Hofer norm ||Φ||, displacing the image of
F −1 [−S, 0).
Then, our original Hamiltonian vector field XF has an orbit homoclinic to x0,
of action bounded above by ||Φ||/(1 − e −S/2 ).
Remark 3.1.3. The condition that dF [Y ] ≤ 1 is so that we may use a change in
F to estimate the time it takes to flow from one point to another with the flow
of Y . Indeed, if F (p) = a and F (q) = b > a are on the same flow line for Y ,
then it will take at least time b − a to go from p to q. This condition normalizes
the scaling of F .
In a certain number of cases, we may immediately conclude that the hypotheses
of Theorem 3.1.2 are verified. The first such case is in R 2n . In this
case, every set with finite diameter in R 2n is Hamiltonian displaceable. This
recovers a result due to Séré, [36, Theorem 1.1]. We recall that a hypersurface
in a symplectic manifold is of restricted contact-type if there exists a global
Liouville vector field transverse to the hypersurface.
Corollary 3.1.4 (Séré). Suppose H : R 2n → R is a smooth Hamiltonian function,
so that :
• The level set H −1 (0) is compact
• H has no critical points on H −1 \ {0}
• dH(0) = 0, d 2 H(0) is non-degenerate and the linearization of XH at zero
is hyperbolic.
• the level set H −1 (0) \ {0} is of restricted contact–type.
25

Then, there is a trajectory of XH homoclinic to 0.
Proof. The fact that M is of restricted contact-type gives us the existence of a
global primitive of ω, which restricts to the contact form on M. This removes the
need for the embedding of M into R 2n to induce an injection on the fundamental
groups.
The next corollary uses a result of Biran and Cieliebak [5, Lemma 3.2].
Suppose (W, ω, Y, F ) is a subcritical Weinstein filling. Denote by M = ∂W , its
positive boundary (we recall a Weinstein filling has no negative boundary). We
may now construct ( ˜ W , ˜ω, ˜ Y , ˜ F ), a (non-compact) subcritical Weinstein manifold
for which the flow of Y exists for all time by gluing on the symplectization
R×M to the boundary of W and extending Y, F and ω over this. This construction
is called the completion of (W, ω, Y, F ). The result of Biran and Cieliebak
may now be stated as follows : Suppose K is a compact set in ˜ W . Then, K is
Hamiltonian displaceable in ˜ W .
From this, we obtain :
Corollary 3.1.5. Suppose (W, ω, Y, F ) is a Weinstein filling with a unique critical
point of index n, which we label x0. We normalize F by the condition that
F (x0) = 0. Then, the level F = 0 carries an orbit homoclinic to x0. (In other
words, we require that F ≤ −δ is a subcritical Weinstein filling for all δ > 0.)
Proof. We then take W0 = F −1 (−ɛ, ɛ) for ɛ > 0 small. The result now follows
from the fact that F −1 (−ɛ) is the positive boundary of a subcritical filling and
is thus Hamiltonian displaceable in the completion of F −1 (−∞, −ɛ].
Finally, the proof of Theorem 3.1.2 leads us to a slight generalization :
Corollary 3.1.6. Suppose (W0, ω0, Y, F ) and (W, ω) as in Theorem 3.1.2, but
so that F has N index n critical points on the level F = 0, again satisfying the
conditions of Theorem 3.1.2.
Then the F = 0 level admits a homoclinic chain. Furthermore, the action of
the homoclinic chain is bounded by N times the bound in Theorem 3.1.2.
We will present the proof of this corollary after we prove the main result,
since the result is actually a corollary of the proof of Theorem 3.1.2.
We recall our definition of a Weinstein domain from the previous chapter.
A Weinstein domain (W, ω, Y, F ) consists of the following data :
• A symplectic manifold (W, ω)
26

• A vector–field Y on W with the property that LY ω = ω (i.e. Y is a
Liouville vector field)
• A Morse function F with the property that Y is gradient–like for F .
We also take F = const on the boundary of W , and that the boundaries are
regular levels of F . We call such a domain critical if at least one of the critical
points has index n = 1 dim W , and subcritical if all of the critical points have
2
index k < n = 1 dim W .
2
We are particularly interested in the critical case. We suppose that there
is only one such critical point, which we will call x0. We will normalize by
assuming that F (x0) = 0. Furthermore, we assume that there are no other
critical points with F ≥ −1.
We consider the critical level M := F −1 (0) \ {x0}. This is a submanifold
of W of contact–type, so it has a preferred contact form λ = iY ω. We wish
to study the Reeb orbits which start and end on the singular point — we will
call such orbits, homoclinics. Indeed, these correspond to homoclinic orbits of
any Hamiltonian system admitting M ∪ {x0} as a (critical) energy level (with
unique critical point at x0) — for instance, F . We note that since Reeb chords
are parametrized by action ( x∗λ), and since homoclinics will have finite action,
these Reeb orbits only exist for finite time.
The proof of the result is very similar to the proof of a result by Mohnke [34],
in which he proves the existence of Legendrian chords, also sometimes called
Reeb chords, and provides a similar bound on their length. In his case, we
are given a Legendrian knot (or other, more complicated Legendrian, in higher
dimensions) ℓ, and ask the question as to whether there exists a trajectory of
the Reeb vector field that intersects the Legendrian twice.
As we discussed previously, a homoclinic orbit is, in some sense, a Legendrian
chord for a singular Legendrian. As we will see below, in a suitable sense, the
“unit” stable and unstable manifolds of an index n critical point are Legendrian
submanifolds. A homoclinic orbit is then a Legendrian chord from the unit
unstable manifold to the unit stable manifold. We will make this idea more
precise in our proof of the result.
Mohnke’s result may be stated as follows:
Theorem 3.1.7 (Mohnke). Let (M, ξ) be a contact manifold with contact form
λ. Let L be a Legendrian submanifold of M.
Suppose that there exists a symplectic manifold (W, ω), symplectically aspherical,
and of bounded geometry (in the sense of Gromov) and S > 0 so that we
may symplectically embed ([−S, 0] × M, d(e s λ)) ↩→ (W, ω) and furthermore, this
embedding induces an injection on the fundamental groups.
If there exists a Hamiltonian diffeomorphism Φ of W with compact support and
27

finite Hofer norm ||Φ|| displacing the image of [−S, 0] × M, then there exists a
Legendrian chord of action bounded above by ||Φ||/(1 − e −S )
The idea of the proof is to construct an immersed Lagrangian submanifold
of [−S, 0] × M, depending on a parameter T , so the Lagrangian is embedded if
and only if no Reeb chord has action less than T . Furthermore, by exploiting
the fact that this Lagrangian lies in a neighbourhood which is a piece of the
symplectization of M, we can show that the area of any disk with boundary
in our Lagrangian has symplectic area equal to an integer times a constant
depending explicitly on T .
If the Lagrangian is embedded, a result of Chekanov (Theorem 3.2.1, below),
together with our assumption on the existence of our Hamiltonian diffeomorphism,
gives the existence of a non–constant holomorphic disk (in W ) with
boundary in the Lagrangian, of symplectic area no greater than ||Φ||. By the
assumption on the induced injection on fundamental groups, this means there
is a disk in [S, 0] × M with boundary on the Lagrangian. By the symplectically
aspherical condition on W , this must have the same symplectic area as
the holomorphic one. We then obtain an inequality giving an upper bound on
T .
Our result on homoclinic orbits will use a similar argument, except this
time, the failure of our Lagrangian to be embedded will force the existence of a
homoclinic orbit.
3.2 Proof of our result
We will now present the proof of theorem 3.1.2. The argument will come in
three steps : the first step is to show that there is a suitable sense in which the
unit unstable manifold of our critical point is a Legendrian. The second step
will be to assume that we have no homoclinic of action less than or equal to T ,
and use this assumption to construct an embedded Lagrangian. Finally, we use
Legendrian surgery to arrange the problem so that we may apply the following
theorem of Chekanov, proved in [7] :
Theorem 3.2.1. Let (W, ω) be a symplectic manifold, either closed or of
bounded geometry (in the sense of Gromov). We denote by J the set of all
almost complex structures J on W , tamed by ω and of bounded geometry. Let
L be a closed Lagrangian submanifold of W . For each J ∈ J , let σ S 2(J) be the
minimal symplectic area of a non-constant J-holomorphic sphere and σD(J) be
the minimal symplectic area of a non-constant J-holomorphic disk with bound-
ary in L. Let now
σ = sup min{σS2(J), σD(J)}.
J∈J
28

Suppose φ : W → W is a Hamiltonian diffeomorphism whose Hofer norm
satisfies
||φ|| < σ.
If φ(L) intersects L transversely then
#L ∩ φ(L) ≥ dim H∗(L, Z2).
Proof of Theorem 3.1.2. Our first step is the construction of the Lagrangian
submanifold. We have assumed that XF has a hyperbolic rest point at x0.
Hence, we have the existence of a global stable and unstable manifold. Let us
denote the stable and unstable manifolds as W s and W u , respectively. Furthermore,
we have that each of these two manifolds has dimension n. We consider
now the linearized flow. In the linear case, the stable and unstable subspaces
are isotropic subspaces. Hence, W u and W s are Lagrangian.
In order to carry out the construction, we need to show that there is a a
suitable notion of a “unit unstable manifold” that can be arranged to be a
Legendrian sphere. In other words, we need the following lemma :
Lemma 3.2.2. Suppose (W 2n , ω, Y, F ) is a Weinstein domain with critical point
x0 on the level F = 0. Suppose XF (0) is a hyperbolic zero. Let W s and W u
denote the stable and unstable manifolds for the flow of XF . Then, for any
neighbourhood of x0, there exists a Legendrian embedding of S n−1 into W u [
W s ], everywhere transverse to XH, with image in the chosen neighbourhood.
Furthermore, for any constant c sufficiently small, we may arrange the Legendrian
so that for any path γ connecting x0 to any point on the Legendrian
sphere, the path has action γ ∗ λ = c.
Proof of Lemma 3.2.2. By the Stable/Unstable Manifold Theorem, we have an
immersion p : Rn → W u , with p(0) = x0. Since W u is Lagrangian, we have
p∗ω = 0. We let λ := iY ω. Thus, d(p∗λ) = p∗ω = 0. Since we are working
in Rn , we have that p∗λ = dG for some function G : Rn → R. The vector–
field XF is tangent to W u , and so we may pull XF |W u back to Rn using p−1 .
We will write ZF = p −1 ∗XF |W
u. We then obtain that G is non-constant since
dG[ZF ] = λ(XF ) = ω(Y, XF ) = dF [Y ] ≥ 0. We recall that dF [Y ] > 0 away
from x0. Thus, G has no critical points away from 0, and level sets of G are
transverse to ZF . We normalize G by the condition that G(0) = 0. Thus, we
have G(x) ≥ 0 with G = 0 only at 0.
We now claim that p(G −1 (const)) is a Legendrian sphere in the unstable
manifold, transverse to XF .
29

By the Stable/Unstable manifold theorem, there is a positive quadratic form
N(x) on R n so that in a neighbourhood of 0, the norm is increasing along orbits
of Z. We consider the sphere S := {N(x) = δ0} for δ0 sufficiently small. Let c
be the minimum of G on this sphere. Then c > 0 and G −1 (c) is contained in
{x |, N(x) ≤ δ0} and thus is compact. Furthermore, G −1 (c) is diffeomorphic to
S, where the diffeomorphism is given by following the flow lines of Z. We then
see that G −1 (c) is a sphere for c > 0 sufficiently small. (Then, it follows that
all level sets of G are spheres.)
We now have that ℓ := p(G −1 (c)) is a Legendrian sphere in the unstable
manifold, transverse to XF . The action of any path γ between x ∈ ℓ and x0 is
then given by G(p −1 (x)) = c.
Let us denote the unit unstable Legendrian given by Lemma 3.2.2 by ℓ. Let
us choose a T1 > 0 for the action necessary to get to ℓ from x0.
At this point, it is more convenient to work with the Reeb vector field than
with the Hamiltonian vector field. In this case, the Reeb vector field is given
by :
X =
1
ω(Y, XH) XH.
Observe that X blows up at x0, and so it is defined on a non–compact space,
F −1 (0) \ {x0}.
Let T > 0. Assume that the orbits of the Reeb vector field with initial data
on ℓ exist for time T . Observe that this is implied by the stronger assumption
that there are no homoclinic orbits (i.e. no orbit starting at ℓ intersecting the
stable manifold). Let φt be the flow associated to the Reeb vector field, X.
Thus, the image of [0, T ] × ℓ by (t, x) ↦→ φt(x) is compact in F −1 (0) \ {x0}.
We may then take a small neighbourhood U ⊂ W0 of x0 whose closure is disjoint
from the image of [0, T ] × ℓ.
We will now construct a new contact manifold, N, realized as a hypersurface
in W0, so that there is a neighbourhood U ′ so that N \ U ′ = F −1 (0) \ U. The
essential idea is that we will see F −1 (0) as the critical level in a Legendrian
surgery, where the surgery has support entirely in the neighbourhood U. Then,
since the Reeb orbit through any initial data on ℓ up to time t avoids U, we may
substitute N for F −1 (0) without any loss of relevant information. (See Figure
3.1.)
To construct N, we will consider a new Weinstein domain (W0, ω0, Y, ˜ F )
obtained by perturbing F inside of U. We wish to take ˜ F : W0 → R satisfying
the following properties :
• ˜ F ≥ F on all of W0.
• ˜ F = F on W0 \ U.
30

• Y is gradient-like for ˜ F .
• ˜ F has no critical points for ˜ F = 0.
Then, if we consider the level set N := { ˜ F = 0}, this will lie entirely in
F −1 [−δ, 0] \ U, where δ > 0 is small (and given by our construction). Furthermore,
N will be a smooth contact manifold, with contact form given by
iY ω0. The corresponding Reeb vector field X ′ will agree with X on N \ U. Let
φ ′ t be the flow of X ′ . Thus, we have that the map
[0, T ] × ℓ → N ⊂ W0
(t, x) ↦→ φt(x)
is an embedding whose image does not intersect the closure of U.
An illustration of the perturbation ˜ F is given in figure 3.1. An example of
such a perturbation can be constructed as follows. Since x0 is a Morse critical
point for F of index n, we may find a neighbourhood of x0 in W0 and coordinates
ξ1, . . . , ξn and ζ1, . . . , ζn on this neighbourhood so that F = n
i=1 ξ2 i − ζ 2 i . Let
β : [0, 1] → [0, 1] so that β(r) = 1 for 0 ≤ r ≤ 1/4 and β(r) = 0 for 3/4 ≤ r ≤ 1,
and −4 ≤ ∂
∂r β(r) ≤ 0. Let δ0 > 0 be small enough so that these coordinates are
defined on a ball of radius δ0 and furthermore, so that their image is entirely in
the neighbourhood U of x0. We set
P :=
1
β
δ2 (ξ
0
2 1 + · · · + ξ 2 n + ζ 2 1 + · · · + ζ 2
n)
˜F := F + ɛP,
where ɛ > 0 will be chosen later in order to satisfy the properties we need.
Then, ˜ F ≥ F , and ˜ F = F outside of U. Furthermore, we check that if ɛ < δ 2 0/4,
we do not introduce any new critical points for ˜ F , and thus 0 is not a critical
value of ˜ F . Finally, we need to check that Y is gradient-like for ˜ F . We observe
that d ˜ F [Y ] = df[Y ] + ɛ dP [Y ]. We have that dP has support in a compact
region given by
1
4 δ2 0 ≤ ξ 2 1 + · · · + ξ 2 n + ζ 2 1 + · · · + ζ 2 n ≤ 3
4 δ2 0.
In this region, df[Y ] is bounded away from zero. We may thus take ɛ sufficiently
small so that d ˜ F [Y ] is bounded away from zero on this region. It then follows
that Y is gradient like for ˜ F .
We let N = ˜ F −1 (0). This is now a smooth, compact contact manifold. We
may symplectically embed [−S/2, 0] × N in (W, ω) by factoring through the
31

ℓ u
x0
ℓ u
F = 0
˜F = 0
F = 1
F = −1
Figure 3.1: The level sets F = 0 and ˜ F = 0. The two level sets agree away
from x0. The zero level of ˜ F and of F are represented in dotted and solid lines,
respectively.
Weinstein domain (W0, ω0). We note that since the image of F −1 [−S, 0] was
displaced by Φ, we have that the image of [−S/2, 0] × N is also.
We now have, by construction, a Legendrian ℓ in N with no Legendrian chord
of action less than or equal to T . Indeed, as ℓ lies in the unstable manifold of
x0, any chord must run through the surgered region. By assumption, this does
not happen.
For the sake of completeness, we will now provide the construction of the
Lagrangian, due to Mohnke. We could, however, conclude immediately by citing
Theorem 3.1.7, and deriving a contradiction.
Our step now is to construct the Lagrangian submanifold (see Figure 3.2).
We first embed
¯q : [0, T ] × [−S/2, 0] × ℓ ↩→ [−S/2, 0] × N
(t, s, x) ↦→ (s, φt(x))
where φt is the flow of the Reeb vector field. This is clearly an immersion. It
is an embedding by our construction, which guarantees no chords and hence no
self-intersection points. We also have that [−S/2, 0] × N ↩→ W0. Let q be the
composition of these two embeddings, so q : [0, T ] × [−S/2, 0] × ℓ ↩→ W0.
We now have q ∗ (e s λ) = e s dt and q ∗ ω0 = e s ds ∧ dt. Fix δ > 0. We find a
circle C that approximates the boundary of [0, T ] × [−S/2, 0]. We may choose
32

s
this circle so that
T (1 − e −S/2
) − δ ≤
L
Figure 3.2: The Lagrangian L.
W u
e
C
s dt ≤ T (1 − e −S/2 ).
Let us call the action of this circle, C =
C es dt.
We now introduce our Lagrangian L = C × ℓ. We observe that any disk
v : D → W0 with boundary in L has symplectic area
D v∗ω =
∂D v∗ (esλ). This is then given by kC, where k ∈ Z is the winding number of v|∂D around
the non-trivial circle C.
We now use the deep result of Chekanov, Theorem 3.2.1. Let J be an almost
complex structure on W , tamed by ω and of bounded geometry. Then, since
dim H∗(L, Z2) > 0 and Φ displaces L from itself, we have the existence of a nonconstant
pseudoholomorphic disk u with boundary in L so that its symplectic
area u∗ω ≤ ||Φ||, the Hofer norm of Φ.
By the fact that the embedding N induces an injection on fundamental
groups, there is a disk u ′ lying entirely in the image of [−S/2, 0] × N with
the same boundary as u. Furthermore, as W is symplectically aspherical, the
symplectic areas u ′∗ω = u∗ω agree. By our construction of L, however, we
have that u ′∗ω = kC for some integer k. The pseudoholomorphic curve u has
positive symplectic area so k ≥ 1. Thus, we have the inequality
C ≤ u ′∗ ω ≤ ||Φ||.
Since δ > 0 was arbitrary, we have (1 − e −S/2 )T ≤ ||Φ||.
Thus, we must have a homoclinic orbit of action no greater than T1 +
||Φ||/(1 − e −S/2 ). As T1 was arbitrarily small, the result now follows.
We will now prove the corollary 3.1.6. This is merely using the previous
argument together with a pigeonhole principle to get a homoclinic chain.
33

Proof of Corollary 3.1.6. Let us label the critical points xi, for i = 1, . . . , N.
For each one, we construct a unstable manifold Legendrian, as in Lemma 3.2.2.
We will denote this by ℓi. Let B denote the bound from the theorem : B =
||Φ||/(1 − e −S/2 ).
We begin with some xi. The argument used in the proof of Theorem 3.1.2
allows us to conclude that there is either a homoclinic orbit of action no greater
than B or that there is an orbit starting on ℓi, asymptotic to some ℓj, of action
no greater than B.
We thus conclude that from each critical point, there is either a homoclinic
orbit of action bounded by B or a heteroclinic orbit of action bounded by B. By
the pigeonhole principle, it follows that there is at least one homoclinic chain,
of length no greater than N. We then obtain that the action of the chain is
bounded above by NB as claimed.
3.3 Applications to the Weinstein conjecture
Surprisingly, Mohnke’s result has some important consequences for the existence
of periodic orbits. In particular, this provides an alternate proof for many special
cases of a result due to Liu and Tian in [31].
Theorem 3.3.1. Let (M, ξ) be a contact manifold with contact form λ.
Suppose that there exists a symplectic manifold (W, ω), symplectically aspherical,
of bounded geometry (in the sense of Gromov) and S > 0 so that we may
symplectically embed ι : ([−S, S] × M, d(e s λ)) ↩→ (W, ω) and furthermore, this
embedding induces an injection on the fundamental groups.
If there exists a Hamiltonian diffeomorphism Φ of W with compact support and
finite Hofer norm ||Φ|| displacing the image of [−S, S] × M, then there exists a
periodic orbit of action bounded above by 4||Φ||/(e S − e −S ).
This implies, in particular, a result that overlaps with Liu and Tian’s result
about the “stabilized” Weinstein conjecture.
Theorem 3.3.2 (Liu–Tian [31, Theorem 1.3]). Let (W, ω) be a symplectic manifold
(either closed, or of bounded geometry). Then, there exists a non-negative
integer l0 so that for any l ≥ l0, any compact, contact-type separating hypersurface
M of (W ⊕ C l , ω ⊕ ω0) carries a closed Reeb orbit.
This result follows as a corollary of Theorem 3.3.1 if the symplectic form on
(W, ω) is exact. (This, in particular, requires that W be non-compact.) In this
case, we may take l0 = 1.
34

This result also follows as a corollary of Theorem 3.3.1 if W is symplectically
aspherical and the embedding of M into W ⊕ C l induces an injection on the
fundamental groups. Again, in this case, we may take l0 = 1.
We also obtain that there exists a symplectic structure on W ⊕ C l so that
any contact-type hypersurface carries a closed Reeb orbit. Indeed, this follows
by a result of Eliashberg :
Theorem 3.3.3 (Theorem 1.3.1 in [12]). Suppose (X 2n , J) is an almost complex
manifold, with n > 2. If X admits an exhausting Morse function F : X → R,
with each critical point of index no greater than n, then there exists an integrable
complex structure J ′ , homotopic to J through almost complex structures, so that
(X, J ′ , F ) is Stein.
We recall that a Morse function F : X → R is said to be exhausting if F
is proper and bounded from below. We say that (X, J ′ , F ) is a Stein domain
if J ′ is an integrable complex structure, F : X → R is an exhausting Morse
function and F is plurisubharmonic with respect to J ′ . From this, we obtain a
symplectic form by ωF := − d( dF ◦ J ′ ).
We put a Morse function H on W . Let N ≤ 2n be the maximal index of its
critical points. Let k = 0 if N ≤ n, and k = N − n otherwise.
We take l > k. Then there exists a Morse function ˆ H : W ⊕ C l → R whose
critical points all have index strictly less than n+l. (We may take ˆ H = H+|z| 2 .)
By Eliashberg’s theorem, it then follows that there exists a J ′ on W ′ for
which F is plurisubharmonic. Let Ω = − d d C F be the corresponding symplectic
form on W ′ . This gives us a subcritical Weinstein domain. This establishes the
claim.
Proof of Theorem 3.3.1. The key point in the proof of this result is to transform
the periodic orbit problem to a Reeb chord problem. We will then use Theorem
3.1.7 to conclude. We note that we have changed the piece of symplectization
we use from [−S, 0] to [−S, S]. This is merely a notational convenience.
In the following, we continue the notation from the previous sections, and
assume that M and W satisfy the hypotheses of Theorem 3.3.1. Let Y = ι∗ ∂
∂s .
This is a (local) vector field on W , defined on the image of [−S, 0] × M. We
note that this is a Liouville vector field.
Given our symplectic manifold, W , we can construct a larger one W, which
admits W as a Lagrangian submanifold. Let W = W × W with the symplectic
form Ω = ω ⊕ −ω. Then, embedding W as the diagonal gives us a Lagrangian
embedding. We note in general that the graph of any symplectomorphism
W → W is a Lagrangian embedding of W into W.
We now have that ι ⊕ ι induces a symplectic embedding of
ι ⊕ ι : [−S, 0] × M × [−S, 0] × M ↩→ W.
35

The symplectic form on the domain, [−S, 0] × M × [−S, 0] × M, is given by
We now observe that Z := ∂
∂s1
Ω0 = d(e s1 λ) ⊕ − d(e s2 λ).
+ ∂
∂s2
is a Liouville vector field for this
symplectic form. This is transverse to the level sets of (s1, x, s2, y) ↦→ s1 + s2.
Thus, we may find an embedding of M := [−S/2, S/2] × M × M into R × M ×
R × M as a contact-type hypersurface by :
ϱ : M → R × M × R × M
(s, x, y) ↦→ (s, x, −s, y)
Thus, M becomes a contact manifold with contact form Λ := ϱ ∗ iZΩ0. We have
and thus
iZΩ0 = e s1 λ ⊕ −e s2 λ
Λ = e s λ ⊕ −e s λ.
We observe that M is not compact. This will not affect our argument, as we
only need a “germ” of a submanifold near (0, x, 0, y).
Let R be the Reeb vector field on M corresponding to Λ. We recall that
X is the Reeb vector field on M corresponding to the original form λ. The
associated Reeb vector field is then given by
Rs,x,y = 1
2 e−s Xx − 1
2 es Xy.
In particular, we observe that the submanifolds of s = const are invariant under
the flow of R. In particular then, while M is not compact, only the compact
subset {s0} × M × M is of relevance in studying the Reeb flow. We also note
that for s = 0, R = X ⊕ −X.
We now have a symplectic embedding of [−S/2, S/2] × M into W by composing
the various embeddings we have introduced so far. We first have
[−S/2, S/2] × M → [−S, S] × M × [−S, S] × M
(τ, s, x, y) ↦→ (τ + s, x, τ − s, y).
Now, we also have that [−S, S] × M × [−S, S] × M symplectically embeds in W
by using ι ⊕ ι. We have that the image of [−S/2, S/2] × M is contained in the
image of [−S, S] × M × [−S, S] × M in W. This also allows us to conclude that
the image of this embedding is Hamiltonian displaceable. Indeed, if Φ is our
Hamiltonian diffeomorphism on W , then Φ⊕Φ is a Hamiltonian diffeomorphism
36

on W. This new Hamiltonian diffeomorphism may be generated by Ht + Ht,
where Ht generates Φ. Thus its Hofer norm, ||Φ ⊕ Φ|| ≤ 2||Φ||.
We also have a Legendrian embedding of M into M by x ↦→ (0, x, x). We
are interested in Legendrian chords for this Legendrian. Indeed, suppose γ :
[0, T ] → M is such a Legendrian chord, of action T . Then, writing γ(t) =
(0, x(t), y(t)),we have ˙x(t) = X and ˙y(t) = −X. Furthermore, since the orbit
starts and ends on M, γ(0) = (0, x0, x0) and γ(T ) = (0, y0, y0). From this, we
obtain that x(t) is an orbit of X that starts at x0 and ends at y0 and that
y(T − t) is an orbit starting at y0 and ending at x0. It follows then that the
orbit through x0 is a periodic orbit of action 2T .
While the contact–type hypersurface we construct is not compact, the Lagrangian
we construct as in the proof of Theorem 3.1.2 lies in a compact region
(independent of T ). We also have that the Lagrangian will always lie in a Hamiltonian
displaceable region. We now apply the result of Mohnke, Theorem 3.1.7,
in order to conclude the existence of periodic orbits of action no greater than
4||Φ||/(e S − e −S )
37

Chapter 4
Pseudoholomorphic curves in
the singular level
4.1 Overview
In this chapter, we will study properties of pseudoholomorphic curves into the
symplectization of a critical level in a four dimensional Weinstein domain.
We consider (W 4 , ω, Y, F ), where (W, ω) is a symplectic manifold (with
boundary), Y is a Liouville vector field and F is a Morse function F : W → R.
We have that Y is gradient-like for F . We also assume that F has a unique
critical point x0 of index 2. We normalize F by F (x0) = 0.
In this chapter, we will first study the critical level in the Weinstein domain,
F −1 (0). After deleting the critical point x0, the critical level becomes a
non-compact contact manifold. The contact form coming from the Weinstein
domain, ıY ω, becomes singular at this critical point, as does the associated
Reeb vector field. Instead of working in this setting, we introduce a compactification
of the critical level by gluing in a Legendrian knot. The contact structure
will extend smoothly across the compactification, but the contact form will not
extend as a contact form.
With certain technical assumptions on the Morse function F in the Weinstein
domain, we will have a neighbourhood theorem for the singularity. We
obtain that a deleted neighbourhood of the singularity double-covers a deleted
neighbourhood of a standard Legendrian knot. We will then define a suitable
class of (singular) almost complex structures, compatible with this double cover.
From this class of almost complex structures, we are led to study two different
Cauchy-Riemann type first order elliptic PDE problems. We will call these
type (I) pseudoholomorphic curves and type (II) pseudoholomorphic curves. We
will show that under certain conditions, type (I) curves are actually instances
38

of type (II) curves. Furthermore, we will show that type (II) pseudoholomorphic
curves satisfy many nice properties relevant to the study of Hamiltonian
dynamics. Specifically, a type (II) finite energy half-plane is asymptotic to a
homoclinic orbit (Theorem 4.4.4). We will also prove that if the asymptotic
limit is a transverse homoclinic orbit, then the convergence is at an exponential
rate (Theorem 4.4.12).
This result, both in its statement and in its method of proof, is quite similar
to analogous facts about punctured pseudoholomorphic curves into symplectizations
of smooth contact manifolds (as in [21, 2, 6]).
In a final section, we discuss the directions of future work indicated by the
results of this chapter. Of particular interest is the development of a contact
homology–like theory generated by homoclinic orbits. Then, the differential
in this theory would be given by counting pseudoholomorphic curves of type
(II). This introduces a very interesting structure on the Floer complex : just
as transverse homoclinic orbits may be glued, so too pseudoholomorphic halfplanes
may be glued. This will introduce a new operation on the Floer complex,
unlike any previously studied. Additionally, a good understanding of the relationship
between type (I) and type (II) curves is needed. This would involve a
generalization of Gromov’s removal of singularities theorem. Finally, a complete
description of the compactification of these spaces is needed (analogous to the
description in [6]).
Ultimately, we wish to understand the relationship between homoclinic orbits
on the singular level and periodic orbits on nearby regular levels, and thus
the relationship between the curves we study here and the ones into the symplectizations
of smooth contact manifolds.
4.2 A compactification of the singular level
4.2.1 Construction of the compactification
We recall that we are working in a Weinstein domain, (W 4 , ω, Y, F ), with positive
and negative boundary ∂W = M + ⊔ M − . We assume that F has a unique
critical point x0, with F (x0) = 0, and of Morse index 2. We are interested in
orbits of the Hamiltonian vector field XF homoclinic to x0. In order to study
these, we will study orbits of the Reeb vector field X = 1
dF [Y ] XF . The orbits of
the two vector fields will be the same up to a time re-parametrization.
We recall, that by definition, a homoclinic orbit γ(t) : R → F −1 (0) ⊂ W is a
trajectory for XF that satisfies limt→±∞ γ(t) = x0. However, we note that this
same orbit, when re-parametrized so that γ(τ) is an orbit for X, has existence
only on a finite τ interval. Indeed, X is singular at x0 (it has a singularity of
39

order 1/r). Thus, for us, a homoclinic orbit is an orbit that exists for a finite
time interval. This is to be expected, since the Reeb vector field parametrizes
its orbits by action, x ∗ λ, and the action of a homoclinic orbit is finite.
In our Weinstein domain, (W 4 , ω, Y, F ), we recall that F has a unique critical
point, x0 ∈ W , with F (x0) = 0. We suppose that this critical point has Morse
index 2 and that x0 is a hyperbolic zero of Y . We further normalize F by
∂ − W = F −1 (−1) =: M − and ∂ + W = F −1 (1) =: M + . These boundaries, M ± ,
are smooth contact manifolds with contact forms
λ ± 0 := iY ω|M ±.
We are interested in studying the qualitative behaviour of the Hamiltonian
vector field XF on the critical level, F −1 (0). We observe that
M sing := F −1 (0) \ {x0}
is a smooth, non-compact, contact manifold with contact form
λ sing = iY ω| M sing.
We recall a key observation from Chapter 2. The flow of Y defines a contactomorphism
between level sets of F . Indeed, if c, c ′ are both positive or both
negative, then by following the flow lines of Y , one obtains a contactomorphism
from F −1 (c) to F −1 (c ′ ). Let W u and W s be the unstable and stable manifolds
of x0 with respect to the flow of Y . Let ψt be the flow of Y . Then, as discussed
in Chapter 2, W u and W s are Lagrangian submanifolds of W . They intersect
regular levels F −1 (c), c = 0, in Legendrian submanifolds, which we denote by
Lc . We observe now that for any c and c ′ , F −1 (c) \ Lc and F −1 (c ′ ) \ Lc′ are
diffeomorphic by means of following the flow lines of Y . If we denote this diffeomorphism
by ψ, we have ψ∗λ = et(p) λ, where t(p) is the time it takes to flow from
p ∈ F −1 (c) to the level F −1 (c ′ ). In particular then, ψ is a contactomorphism.
We use this construction to obtain a contactomorphism between the smooth
part of the singular level M sing = F −1 (0) \ {x0} and M ± \ L ±1 . We let t :
(M ± \ L ± ) → R associate to a point p, the time it takes to flow from p to the
level F = 0. The diffeomorphism induced by the flow of Y pulls the contact
form on the singular level λsing back to the contact forms
We let
λ ± = e t(p) λ ± 0 on M ± \ L ± .
f + := e t(p) |M + and f − = e t(p) |M −.
This construction allows us identify M sing with an open subset of a smooth,
closed contact manifold. Indeed, the contactomorphism induced by the flow of
40

Y identifies M sing with M + \ L +1 or with M − \ L −1 . We may then compactify
M sing by gluing in the missing Legendrian knot, L ±1 . We then have that M ±
are two possible compactifications of M sing . The price we pay is that we no
longer have a contact form defined on all of M ± . Instead, we obtain a contact
form that is singular along L ±1 .
We observe that the diffeomorphism induced by the flow of Y induces a singular
contact form on M ± , but the singularity is of a different nature depending
on whether we consider M + or M − . Indeed, we have that f + extends smoothly
to be 0 along the singular Legendrian L + . However, f − becomes unbounded
as we approach the singular Legendrian. Instead, 1/f − extends smoothly to be
zero along the singular Legendrian L − . In the following analysis, we will need
a notion of bounds on gradients. It will be much more convenient to work with
the compactification to (M + , ξ) with singular contact form λ + = f + λ + 0 . We will
point this out explicitly when we discuss gradient bounds (the work leading to
Proposition 4.4.6).
In the following, we will then consider a smooth contact manifold (M, ξ)
with smooth contact form λ0. We have a Legendrian knot L ∈ M and a smooth
function G : M → [0, ∞) so that G = 0 on L, and G > 0 on M \L. We consider
then the singular contact form λ = Gλ0. This will be the model we use for all
of the subsequent work.
4.2.2 A local model of the singularity
For our analysis, we will need to have a good model of the behaviour of λ = Gλ0
near the Legendrian. In the process, we will place a symmetry condition on the
functions G we allow — this in turn, will represent a condition on the class of
Weinstein domains (W, ω, Y, F ) we consider. For the purpose of understanding
Legendrian surgery, this is not a difficulty, since the standard construction from
Weinstein (as discussed in Section 2.2.2) satisfies our symmetry condition. In
general, we are also interested in studying homoclinic orbits on the critical
level in a given (W, ω, Y, F ). We will show that in the case of a strictly nondegenerate
pair (Y, F ), we may make an arbitrarily small perturbation of F
(small in the C 1 topology), so that the resulting Fɛ satisfies our symmetry
condition, and so F = Fɛ outside a small neighbourhood of the critical point.
Thus, if we can conclude that a homoclinic orbit exists by virtue of the existence
of a pseudoholomorphic curve, then we may use a limiting argument to establish
existence on the critical level for F .
We recall that the pair (Y, F ) is non-degenerate if F is a Morse function, Y is
gradient-like with respect to F and furthermore, the zeros of Y are hyperbolic
and the zeros of XF are hyperbolic. We say the pair (Y, F ) is strictly nondegenerate
if, in a neighbourhood of each critical point p of F , there exists a
41

Darboux chart on (W, ω) and a constant so that Y [F ] ≥ c(|x| 2 + |y| 2 ).
We will now show that if G arises as an f + for a strictly non-degenerate
Liouville pair, then it is very close to being of the form described by the standard
model in Section 2.2.2.
We recall that, as we computed in Section 2.2.2, if we introduce exponential
polar coordinates to map R − × S 1 × S 1 → Nbd(L), we may write
α − = e s (cos θ ds − sin θ dθ + sin θ dφ)
α + = e s/2 (cos θ ds − sin θ dθ + sin θ dφ)
αsing = e 2s/3 (cos θ ds − sin θ dθ + sin θ dφ)
In particular, in these coordinates, G(s, φ, θ) = G(s) = e s/6 .
We will denote by Nδ ⊂ R 2 × S 1 the standard Legendrian neighbourhood,
and αstd = dx+y dφ the standard contact form. Let (s, θ, φ) be the exponential
polar coordinates on Nδ \ Lstd by x = e s cos θ, y = e s sin θ. (Thus, αstd =
e s (cos θ ds + sin θ dφ − sin θ dθ).)
Let us now consider the more general situation of an arbitrary pair (Y, F ),
where the critical points are strictly non-degenerate. Then, by definition, we
may take a Darboux chart in which dF [Y ] ≥ c(|x| 2 + |y| 2 ). Thus, if we write
Y = Y0 + Y1, where Y0 is the linear part of Y , and Y1 is higher order, we obtain
that Y0 is also Liouville (and thus, Y1 is a Hamiltonian vector field). Let us
write F = F0 + F1, where F0 is quadratic and F1 is higher order. We then have
that Y0 is locally gradient-like for both F and F0. We may thus assume that in
our local coordinates, Y = Y0.
Lemma 4.2.1. Suppose Y0 is a linear Liouville vector field on (R 2n , ω0). If
Y0(0) = 0 and has n expanding directions and n contracting directions, then
there exist positive constants b1, b2, . . . , bn so that :
n ∂ ∂
Y (x, y) = (1 + bi)xi − biyi
∂xi ∂yi
i=1
n
= ∇ (1 + bi) 1
1
.
i=1
2 x2i − bi
2 y2 i
(Here, ∇ denotes the gradient with respect to the standard metric on R 2n .)
Proof. By assumption Y0(0) = 0 and the zero is hyperbolic, with n contracting
directions and n expanding directions. Its stable and unstable manifolds are
then Lagrangian submanifolds of R 2n , intersecting transversally at 0.
42

We may now find a symplectomorphism on R2n , fixing 0, so that the stable
manifold is mapped to Rn and the unstable manifold to J0Rn .
Writing
n n
Y0 = a j
i xi
∂
− b
∂xj
j
i yi
∂
∂yj
j=1
i=1
and using the fact that d(iY0ω0) = LY0ω0 = ω0, we compute : a i j = 0 and b i j for
i = j, a i i − b i i = 1. In order to satisfy the requirement that Y0 have the correct
index, we also have that b i i > 0. The result now follows.
Write z = (x, y) ∈ R 2n . Let B be the matrix so that Y (z) = Bz. Let
H(z) = |x| 2 − 1/2|y| 2 . We write F (z) = F0(z) + F1(z), where F0(z) =< Az, z >
for some self-adjoint matrix A, and F1(z) = O(|z| 3 ). By assumption, AB is
positive. Let | · | be the norm < ABz, z > 1/2 . This norm is equivalent to the
standard norm on R 2n .
Let α = ıY ω|{H=0}. Let λ = ıY ω|{F =0}. Let λ ′ = ıY ω|{F0=0}. By following
the orbits of Y , we have that {H = 0} \ {0} is diffeomorphic to {F0 = 0} \ {0}
and to {F = 0} \ {0}. We wish to estimate the amount of time it takes to flow
from {H = 0} \ {0} to {F = 0} \ {0}. To do this, we will first estimate the time
it takes to flow from {H = 0} \ {0} to {F0 = 0} \ {0}. Then, we will estimate
the time it takes to flow from {F0 = 0} \ {0} to {F = 0} \ {0}.
We first notice that {H = 0} \ {0} and {F0 = 0} \ {0} are cones. Suppose
p is a point so that H(p) = 0 and S(p) is the time so that F0(ψS(p)(p)) = 0.
Then, since the flow of Y is linear, we have for any r = 0, F0(ψS(p)(rp)) = 0.
Thus, S(p) = S(p/||p||).
Now, we consider the time it takes to flow from {F0 = 0} \ {0} to {F =
0} \ {0}. We denote this time by T (x) for points x with F0(x) = 0. Let x0 so
that F0(x0) = 0. Let x1 = ψT (x0)(x0) be the point on {F = 0}, and x(t) the
orbit between them. Let y(t) = µx(t) for some µ > 0. We then obtain:
F (y(t)) − F (y(0)) = F (y(t)) − F1(y(0))
=
t
= µ 2
dF [Y ]|µx(t) dt
dF0[Y ]|x(t) + 1
µ dF1(µx(t))[Y
(x(t))] dt .
0
t
0
We now observe that by construction, | dF1(z)| ≤ C|z| 2 , so we obtain :
µ 2
t
dF [Y ]|x(t) − µ
0
3 C ′
t
0
|x(t)| 3 ≤ F (y(t)) − F1(y(0))
43
≤ µ 2
t
dF [Y ]|x(t) + µ
0
3 C ′
t
0
|x(t)| 3

With tµ = T (µx0), we obtain :
µ 2
tµ
0
|x| 2 − µ 3 C ′ |x(t)| 3
≤ −F1(µx0) ≤ µ 2
tµ
0
|x| 2 + µ 3 C ′ |x(t)| 3
.
We restrict our attention to a sufficiently small neighbourhood of 0 so that
C|x| 3 ≤ 1/2|x| 2 . We also consider only those µ ∈ (0, 1). We recall that
|F1(µx0)| ≤ Cµ 3 |x0| 3 .
Then, we obtain :
so we have
thus,
µ 2 tµ
| |x|
0
2 | dt ≤ C ′′ µ 3 |x0| 3
|tµ| inf |x(t)|
0≤t≤T (x0)
2 ≤ C ′′ µ|x0| 2
|tµ| ≤ C(x0)µ.
Thus, if we consider x = e s v, where ||v|| = 1, then we have
T (x) ≤ C(v)e s .
Thus, it takes time τ(p) = S(p) + T (ψS(p)(p)) to flow from a point p in
{H = 0} to the corresponding point q ∈ {F = 0}. This satisfies the decay
estimate (for s < 0):
|τ(e s p) − S(p)| ≤ Ce s ,
This gives us the following result:
Lemma 4.2.2. Suppose that (Y, F ) are a strictly non-degenerate Liouville pair,
defined in a neighbourhood of 0 ∈ R 4 . Suppose furthermore, that Y is linear.
Then, with the local construction and coordinates as in Section 2.2.2, we have
the following estimate :
f + = e s/6+E(s,φ,θ)+B(φ,θ)
where |E(s, φ, θ)| ≤ Ce s for −s sufficiently large.
44

(We recall that the coordinates (s, θ, φ) come from taking exponential polar
coordinates in a deleted tubular neighbourhood of the model Legendrian in the
Legendrian neighbourhood theorem.)
From this, it also becomes clear that we can make a C 1 small perturbation of
F that is supported in an arbitrarily small neighbourhood of L, so that G = f +
becomes only a function of s. To do this, we can make use of an averaging
construction.
We are now able to prove our most important result on the local structure
of the singularity.
Proposition 4.2.3. Let β := cos θ ds+sin θ dψ. This is a contact form on R − ×
S 1 × S 1 . Suppose G : R − × S 1 × S 1 → (0, ∞) with G(s, θ, ψ) = e 2s/3 H(s, θ, ψ),
with H(s, θ, ψ) = H(s, θ, ψ + π) and c ≤ H ≤ 1/c for some constant c > 0.
Suppose also that lims→−∞ H(s, θ, ψ) exists for all θ, and ψ, and is independent
of ψ and θ.
Then, there exists a double covering local diffeomorphism
so that there exists a function
Ψ : R − × S 1 × S 1 → (D 2 \ {(0, 0)}) × S 1
F : D 2 × S 1 → (0, ∞) with
Ψ ∗ F ( dx + y dη) = Gβ.
The smoothness of F depends on how fast H decays to its limit as s → −∞.
Proof. We observe that the following map :
Ψ : (s, θ, ψ) ↦→ (e 2s/3 cos θ, e 2s/3 sin θ, 3θ + 2ψ)
satisfies Ψ ∗ ( dx + y dη) = e 2s/3 β.
We notice that this is a double covering map. In particular, (s, θ, ψ) and
(s, θ, ψ + π) are mapped to the same point. By assumption, G(s, θ, ψ) =
G(s, θ, ψ+π) and so it pushes forward to a function F on (D 2 \{(0, 0)})×S 1 . Our
conditions on H now guarantee that this function F extends across {(0, 0)}×S 1 .
H must have an exponential decay rate for F to become differentiable at 0 (the
higher the decay rate, the more derivatives F has).
This gives us that in our situation, if G is sufficiently symmetric, then a
neighbourhood of the singular Legendrian double covers a neighbourhood of
45

ℓ s
ℓ u
ℓu (a) Slice of the Legendrian neighbourhood in
the singular case, given by φ =const.
ℓ s
Ψ(ℓ s )
Ψ(ℓ u )
(b) Image of the slice under Ψ, in the standard
Legendrian neighbourhood
Figure 4.1: Observe that in the singular case (left), ℓ u and ℓ s are embedded
Legendrians, but they each intersect the slice twice. They each double cover
their image in the standard neighbourhood (right).
a Legendrian with a smooth contact form. In particular, we will take G to
be independent of φ and θ, as in the model case of a Weinstein handle, and
this construction becomes somewhat overkill. We illustrate the double cover in
Figure 4.1.
This double cover allows us to define a new notion of Reeb orbit.
Definition 4.2.4 (Generalized Reeb orbit). We will say that a continuous map
γ : [a, b] → M is an orbit of the Reeb vector field if:
• for t so that γ(t) /∈ L, we have that γ is differentiable with γ ′ (t) = X(γ(t))
and
• for t so that γ(t) ∈ L, we have a neighbourhood (t − δ, t + δ) on which
Ψ(γ(t)) is defined and Ψ(γ(t)) is an orbit of the standard Reeb vector field
∂
∂z .
In particular, this allows for certain homoclinic chains to be viewed as orbits
of the Reeb vector field. This definition gives that γ(t) ∈ L for isolated t.
Finally, we note that this definition destroys uniqueness of orbits through a
given point. Instead, we have local uniqueness for points p /∈ L, but when we
hit L, the orbit has two “choices” for how to leave L.
4.2.3 A local model for the neighbourhood of a homoclinic
orbit
46

From the local model for the singular Legendrian, we may obtain a model for
a neighbourhood of a homoclinic orbit (in the generalized sense of definition
4.2.4).
Definition 4.2.5 (Simple homoclinic orbit). We say that a homoclinic orbit γ
is simple if γ : [0, T ] → M has γ(0) ∈ L and γ(T ) ∈ L but γ(t) /∈ L for all
0 < t < T .
We remark that our definition of Reeb orbit (in our generalized sense) allows
us to have Reeb orbits that intersect L multiple times. The intersection must
occur at discrete times, however. For any homoclinic orbit, we may decompose
it into its simple components.
Now, in order to study the neighbourhood of the homoclinic orbit, we will
need the following lemma we proved earlier :
Lemma 3.2.2. Suppose (W 2n , ω, Y, F ) is a Weinstein domain with critical
point x0 on the level F = 0. Suppose XF (0) is a hyperbolic 0. Let W s and
W u denote the stable and unstable manifolds for the flow of XF . Then, for any
neighbourhood of x0, there exists a Legendrian embedding of S n−1 into W u [
W s ], everywhere transverse to XH, with image in the chosen neighbourhood.
Furthermore, for any constant c sufficiently small, we may arrange the Legendrian
so that for any path γ connecting x0 to any point on the Legendrian
sphere, the path has action γ ∗ λ = c.
Proposition 4.2.6. Let γ : [0, T ] → M be a simple homoclinic orbit for the
(singular) Reeb vector field, with γ(0) ∈ L and γ(T ) ∈ L.
Suppose γ(0) = γ(T ). Then, there exists a neighbourhood U of γ([0, T ]) in
M and neighbourhoods V0 ∋ γ(0) and V1 ∋ γ(T ) and a local diffeomorphism
Φ : U → U ′ ⊂ R 3 so that:
1. Φ(γ(t)) = (0, 0, t).
2. The restriction Φ|U\(V0∪V1) is a diffeomorphism.
3. Φ ∗ αstd = λ.
4. Φ(L ∩ V0) = {(x, 0, 0)} ∩ U ′ for |x| small, and
5. Φ(L ∩ V1) = {(f1(τ), f2(τ), f3(τ)} ∩ U ′ ,
for τ ∈ (−δ, δ), and (f1(0), f2(0), f3(0)) = (0, 0, T )
6. Φ∗ ˜ J is an almost complex structure on R × U ′ adjusted to the standard
contact form α0.
47

In the case that γ(0) = γ(T ), there exists a lift
h : U → Nbd(γ([0, T ])) ⊂ M
so that h ∗ λ = λ ′ , where λ ′ is a singular contact form with all the properties of
λ, defined on U. Then there exists a local diffeomorphism Φ : U → U ′ , as above.
Proof. We will prove the existence of Φ by construction, starting with the existence
of Ψ. In order to simplify the notation, we will only consider the case
γ(0) = γ(T ). The idea carries through identically in the more general case.
To do this construction, we will first consider the unit stable/unstable Legendrian
manifolds given by Lemma 3.2.2. The homoclinic orbit is a Legendrian
chord between these two Legendrian knots. We will use a tubular neighbourhood
theorem in order to construct Φ on a neighbourhood of the orbit between
these two Legendrians. Then, we will use Ψ, the double cover that exists in a
neighbourhood of the singular Legendrian L, to continue Φ into a neighbourhood
of L.
The construction will use the fact that if we have a diffeomorphism F : U ⊂
R 3 → V ⊂ R 3 with F ∗ αstd = αstd, then F (φt(p)) = φt(F (p)) for p ∈ U and
t small enough so that φt(p) ∈ U. Thus, we may extend F to {φt(p) | p ∈ U}
by F (φt(p)) := φtF (p). If U and V are chosen so that no Reeb orbit exits U
or V and then re-enters at a later time, then this is well-defined. In particular,
this is well defined if U and V are convex. Furthermore, with this restriction
on U and V , this defines the extension of F uniquely. Indeed, any two such
diffeomorphisms that agree on U and V must then agree on {φt(p) | p ∈ U}.
Let U be the neighbourhood of L on which Ψ is defined, and let Nδ \ Lstd
be its image. Recall that αstd = dx + y dη and Lstd = {(0, 0, η)}.
Take ℓu and ℓs, unit unstable and stable Legendrian manifolds, as in Lemma
3.2.2, sufficiently close to L that they intersect U. We see that Ψ(ℓu) and Ψ(ℓs)
are given by {(−c, 0, η)} and {(c, 0, η)} respectively, where c > 0 is small.
We have then that p := γ(c) ∈ ℓu and q := γ(T − c) ∈ ℓs. We have that
restricted to a neighbourhood of p, Ψ is a diffeomorphism onto its image. We
then have Ψ −1 : B((0, 0, c), ɛ) ⊂ R 3 → U1, a neighbourhood of p. We have that
(Ψ −1 ) ∗ λ = αstd. We also take ɛ sufficiently small that φT −c(U1) ⊂ U \ L.
We now introduce the map
F1 : Dɛ × [c, T − c] −→ M
(x, y, z) ↦→ φz−c(Ψ −1 (x, y, c)).
This is a diffeomorphism onto its image. Furthermore, we have that F1 ∗ λ = αstd.
We now have that F −1
1 , in a neighbourhood of γ(τ ′ ), maps ℓ s to a Legendrian
arc in Dɛ × [τ, τ ′ ]. Let us denote this arc by ˇ ℓ. Then, there exists
48

a diffeomorphism G : Nbd( ˇ ℓ) → Nbd((0, 0, 0)) so that G( ˇ ℓ) = Nbd((0, 0, 0)) ∩
{(0, 0, η) | |η| small.}. We now extend G to a neighbourhood of {(0, 0)}×[−T, T ]
by using the Reeb flow. In particular, we have that G −1 ◦Ψ is defined on a neigh-
bourhood of γ([T − c, T ]).
Observe that G −1 ◦ Ψ : Nbd(ℓ s ) → R 3 and F −1
1
sufficiently small neighbourhood of ℓ s .
: Nbd(ℓ s ) → R 3 agree on a
We now take our map Φ = Ψ near p, and Φ = F −1
1 in the region in which it is
defined. Finally, we take Φ = G −1 ◦Ψ in a neighbourhood of γ([T −c, T ]). These
maps agree where their domains of definition overlap, so Φ is well defined. By
construction, this is a local diffeomorphism from a neighbourhood of γ([0, T ])
onto its image. Furthermore, it is a diffeomorphism onto its image once a
neighbourhood of L is deleted from its domain of definition.
4.3 Type (I) and (II) ˜ J-holomorphic curves
In this section, we introduce a special class of almost complex structures, compatible
with the double cover Ψ introduced in Prop 4.2.3. We then define the
two classes of pseudoholomorphic curve we will be interested in. Finally, we
point out some cases in which the two classes agree.
4.3.1 The singular almost complex structure
By Proposition 4.2.3, there exist a δmax > 0, and a local diffeomorphism Ψ
defined in a neighbourhood U of L, so that Ψ : (U \ L) → (Nδmax \ Lstd) is a
double cover. Furthermore, we have Ψ ∗ αstd = λ. We will let Uδ = Ψ −1 (Nδ \
Lstd) ∪ L and Mδ = M \ Uδ. We observe then that Mδ is a compact contact
manifold, with convex boundary.
In the following, we will abuse notation and let Ψ denote either be the map
Ψ : U −→ Nδmax
or the map Ψ : R × U −→ R × Nδmax,
by extending Ψ by the identity on the R component.
Our local model of the singularity allows us to define what we mean by an
almost complex structure adjusted to the singular contact form λ.
Definition 4.3.1. We say that ˜ J is an almost complex structure adjusted to
the singular contact form λ = f ± λ0 if ˜ J is adjusted to λ away from L, and
furthermore, T Ψ ◦ ˜ J ◦ ( T Ψ) −1 is a complex structure adjusted to αstd and
extends smoothly across Lstd.
49

We recall that a smooth almost complex structure ˜ J is adjusted to a
(smooth) contact form α if ˜ J is R invariant, ˜ J ∂
∂s = X and ˜ Jξ = ξ with ˜ J|ξ
compatible with the symplectic form dλ. This terminology is as in [6], though
this class of almost complex structures was previously introduced by Hofer in
[21].
In the following, we will refer to an almost complex structure on M \ L,
adjusted to the singular contact form λ as a singular almost complex structure
if it satisfies this Definition 4.3.1.
Remark 4.3.2. The class of almost complex structures defined by 4.3.1 is not
empty. The key observation is that the space of almost complex structures
compatible with a symplectic form is contractible. Then, given any two almost
complex structures adjusted to a given contact form, we may find a path of
almost complex structures between them. In particular, this allows us to do
an “averaging” construction to glue together local constructions. Thus, we
construct an almost complex structure on U. We then take an almost complex
structure on Mδ. We may then adjust the two almost complex structures to
agree on Uδmax \ Uδmax/2 and then obtain an almost complex structure on M
satisfying definition 4.3.1.
To obtain an almost complex structure on U, we start by choosing an almost
complex structure on Nδmax, adjusted to αstd. By means of Ψ, we may pull this
complex structure back to a complex structure on U \ L.
4.3.2 Type (I) pseudoholomorphic curves
The first type of pseudoholomorphic curve we introduce is the most natural.
Suppose ˜ J is an almost complex structure as in Definition 4.3.1. Then we say
that a continuous map ũ : (Σ, j) → R × M is a Type (I) pseudoholomorphic
curve if T ũ(z) ◦ j = ˜ J(u(z)) T ũ(z) for all z ∈ Σ so that u(z) /∈ L.
In particular, we note that if we have a sequence of pseudoholomorphic
curves that are pseudoholomorphic for a sequence of smooth almost complex
structures converging to ˜ J in C ∞ loc
on compact subsets of M \ L, and they
satisfy a uniform gradient bound, then a subsequence will converge to a Type
(I) pseudoholomorphic curve.
We will call the set {z | u(z) ∈ L}, the set of singularities of ũ, and denote
it by Γ(u, L).
Unfortunately, these curves are difficult to deal with. This will lead us to
the second class of pseudoholomorphic curve we consider.
4.3.3 Type (II) pseudoholomorphic curves
50

In light of the double cover we obtained in Proposition 4.2.3, another natural
elliptic PDE to consider is that the curve satisfy the non-linear Cauchy-Riemann
equations when pushed down by the double cover.
Definition 4.3.3. We say that a curve ũ = (a, u) : (Σ, j) → R × M is Type
(II) pseudoholomorphic with respect to the singular almost complex structure
˜J, satisfying the definition 4.3.1 if
• ũ is continuous,
• ũ is pseudoholomorphic on {z | u(z) ∈ L}, and
• Ψ(ũ) is pseudoholomorphic as a map into R × Nδmax, where defined.
As above, we will call the set {z | u(z) ∈ L}, the set of singularities of ũ, and
denote it by Γ(u, L).
We see that a Type (I) pseudoholomorphic curve is automatically a Type (II)
pseudoholomorphic curve. The converse is not known in general. Nevertheless,
we may conclude that in certain circumstances, the two types coincide.
Proposition 4.3.4. Let ũ : (Σ, j) → R × M be a Type (I) pseudoholomorphic
curve. Let Γ ′ be the set of accumulation points of Γ(u, L). If Γ ′ accumulates in
a discrete set, then ũ is a Type (II) pseudoholomorphic curve.
This result is actually a corollary of Gromov’s theorem on the removal of
singularities for a pseudoholomorphic curve :
Theorem 4.3.5. Suppose (W, ω) is a symplectic manifold (either compact or
of bounded geometry) and J is an almost complex structure compatible with ω.
Then, if we have a pseudoholomorphic map u from the punctured, closed disk
of finite symplectic area :
u : ¯
D \ {0} → W with u ∗ ω < ∞
then, u has a smooth (and hence pseudoholomorphic) extension across 0.
(For a proof, we refer the reader to [33].)
We will use two simple corollaries of this result. The first applies this theorem
to the symplectization of a contact manifold. The second discusses what
happens if we have more than a single singularity.
Corollary 4.3.6 (Contact Version). Suppose M is a contact manifold with
contact form λ and ˜ J is an R–invariant almost complex structure on the symplectization,
adjusted to the contact form. Any pseudoholomorphic map from a
punctured disk into the symplectization, ũ = (a, u) : ¯ D \ {0} → R × M with
u ∗ dλ < ∞ and a bounded admits a smooth extension across 0.
51

This follows immediately from estimating the symplectic area of the form
d(e t λ) in terms of the bound on a and the area u ∗ dλ. We may then apply
Gromov’s theorem.
Corollary 4.3.7. Suppose (W, ω) is a symplectic manifold (either compact or
of bounded geometry) and J is an almost complex structure compatible with ω.
Let Γ be a closed subset of the open disk, whose set of accumulation points
is precisely {0}.
Then, if we have a pseudoholomorphic map u from the closed disk, with
singularities at Γ, but with finite symplectic area :
u : ¯
D \ Γ → W with u ∗ ω < ∞
then, u has a smooth (and hence pseudoholomorphic) extension across Γ.
The proof is by iterated application of Gromov’s removal of singularities.
In order to prove the result 4.3.4, we consider Ψ ◦ ũ : U → R × Nδ, where
U ⊂ Σ is the set on which Ψ ◦ u is defined. Since Γ ′ has a discrete set of
accumulation points, we may apply Gromov’s removal of singularities iteratively.
We then have that Ψ◦ũ admits a pseudoholomorphic extension across Γ. Thus,
we have that Ψ ◦ ũ is pseudoholomorphic where defined, and hence ũ is a Type
(II) pseudoholomorphic curve.
4.4 Asymptotic behaviour of a Type (II) finite
energy half-plane
In this section, we prove some results about the asymptotic behaviour of a Type
(II) pseudoholomorphic curve. Of greatest interest is that a Type (II) pseudoholomorphic
half-plane is asymptotic to a homoclinic orbit. The convergence
will be at an exponential rate if the asymptotic limit is transverse.
4.4.1 Convergence to a homoclinic orbit
In this section, we prove that Type (II) ˜ J-holomorphic half-plane, in the sense
of definition 4.3.3, with finite energy, is asymptotic to a homoclinic orbit.
We write H = {z ∈ C | ℜ(z) ≥ 0}.
Theorem 4.4.1. Let M, λ and ˜ J as in Definition 4.3.1. Suppose that ũ : H →
R × M is Type (II) ˜ J holomorphic, in the sense of Definition 4.3.3, with finite
52

energy E(ũ) = T < ∞ and boundary condition u(∂H) ∈ L. Let T = u ∗ dλ.
Then, for any sequence R ′ k → ∞ has a subsequence Rk so that
u(Rke it ) → x(T t/π)
where x(t) is a homoclinic orbit of the (singular) Reeb vector field. The convergence
is in C ∞ ([−π/2, π/2], M).
We are especially interested in the limit if this is a transverse homoclinic
orbit. We will use the term non-degenerate to emphasize the similarity with
non-degenerate periodic orbits, as in Floer theory or Contact Homology, or
non-degenerate Reeb chords (as in [2]).
Definition 4.4.2 (Non-degenerate homoclinic orbit). Let ℓu and ℓs be the unit
unstable and stable Legendrians given by Lemma 3.2.2, of action ɛ > 0 (smaller
than the shortest homoclinic). Let γ be a simple homoclinic orbit. Suppose
γ is parametrized so that γ(0) ∈ ℓu and γ(τ) ∈ ℓs. Then, the orbit γ is nondegenerate
if T γ(τ) T γ(0)ℓu ⊕ T γ(τ)ℓs = ξγ(τ).
Equivalently, the map R×ℓu → M : (t, p) ↦→ φt(p) intersects ℓs transversally
at (τ, γ(0)).
A homoclinic orbit γ is non-degenerate if each of its simple components are.
We remark that this is equivalent to requiring that Φ(γ) be a non-degenerate
Legendrian chord.
In the case that the asymptotic limit is a non-degenerate homoclinic orbit,
we have :
Theorem 4.4.3. Let ũ : H → R × M as in the hypotheses of 4.4.1. If the
asymptotic limit, x : [0, T ] → M is a non-degenerate, simple homoclinic orbit,
then
lim
R→∞ u(Reit ) = x(T t)
where x(t) is a homoclinic orbit of the (singular) Reeb vector field. The convergence
is in C ∞ ([−π/2, π/2], M).
We also establish that a pseudoholomorphic curve with a weaker boundary
condition also converges to a homoclinic orbit :
Theorem 4.4.4. Let M, λ and ˜ J as in Definition 4.3.1. Suppose that ũ : H →
R × M is Type (II) ˜ J holomorphic, in the sense of Definition 4.3.3, with finite
energy E(ũ) < ∞ and boundary condition at infinity
d(u(0 + it), L) → 0 as |t| −→ ∞.
53

Let T = u ∗ dλ. Then, for any sequence R ′ k → ∞ there is a subsequence Rk so
that
u(Rke it ) → x(T t)
where x(t) is a homoclinic orbit of the (singular) Reeb vector field. The convergence
is in C ∞ ([−π/2, π/2], M).
In order to prove Theorem 4.4.1, we must first establish some properties of
Type (II) pseudoholomorphic curves for a singular almost complex structure,
in the sense of Definition 4.3.3. Due to the existence of the double cover, as
in Proposition 4.2.3, we will be able to show that ˜ J-holomorphic curves, for
our singular almost complex structure, share many of the same properties as
pseudoholomorphic curves for a regular almost complex structure.
Our results are all of an analytical nature, and to develop them, we must
introduce various Sobolev spaces. Of greatest importance to us are the W 1,p
spaces, for p > 2. It will be convenient to realize our function spaces as linear
spaces, so we will need a way to embed our problem into a high dimensional
R N . This is where we use the specific compactification of the singular level
to M + with contact form f + λ + 0 . We equip M + with the metric coming from
λ + 0 (specifically g = λ + 0 ⊗ λ + 0 + dλ(·, J·) where J is a suitable almost complex
structure on the contact structure). We now embed this isometrically in R N for
N sufficiently large. By the explicit form of the local diffeomorphism Ψ given
by Proposition 4.2.3, we see that || T Ψ|| ≤ C where it is defined. This will allow
us to obtain gradient bounds “downstairs” if we have them “upstairs”.
Our first result will be to show that gradient bounds imply C ∞ bounds.
We first recall the result for smooth almost complex structures on compact
manifolds. A good reference for the proof is Appendix B of [33]. It is also
proved in Chapter 6 of [4].
Theorem 4.4.5 (Theorem 4.1.1 in [33]). Suppose (W, J) is an almost complex
manifold. We denote by D, the unit disk in C. Assume un : D → W is
a sequence of J-holomorphic curves with a uniform C 1 bound. Then, for all
0 < r < 1, for all p > 2 and for all k ∈ N, there exists a constant C = C(r, p, k)
so that
||ũn|| W k,p (Dr) ≤ C.
In particular, a subsequence converges in C ∞ loc .
From this, we will deduce a comparable result for our singular setting :
Proposition 4.4.6. Let ũn = (an, un) : D → R × M be a sequence of ˜ Jholomorphic
curves for a singular almost complex structure ˜ J, in the sense of
Definition 4.3.1.
54

Suppose the sequence has a uniform gradient bound : |∇ũn| ≤ C and a bound
on an(0) (say, |an(0)| ≤ C ′ for all n). In other words, we require a C 1 bound
on the sequence.
Then, there exists an 0 < rmax ≤ 1, so that for all 0 < r < rmax, for all p > 2
and for all k ∈ N, there exists a C = C(r, p, k, l) so that
||ũn|| W k,p (Dr) ≤ C.
In particular, a subsequence converges in C ∞ loc .
Proof. Our result will follow from Theorem 4.4.5, and from the fact that Ψ
is Lipschitz (as observed above). This uses, in an essential way, the fact that
we are working with the compactification to M + with singular contact form
λ = Gλ0, with G = 0 along L.
Indeed, we introduce rmax = min(1, δmax/4C), where C is the bound on the
gradients of ũn. Then, for any ũ in our sequence, by the bound on ∇ũ, and the
triangle inequality, we have one of the two following cases :
1. The image of u is totally contained in the domain of the double cover :
u(D) ⊂ Uδmax
2. The image of u is totally contained in the Mδmax/2.
In either case, the bound follows immediately from the result 4.4.5.
We will prove Theorems 4.4.1, 4.4.4 by building up the following propositions.
In all of them, we take M, λ and ˜ J as in Definition 4.3.1.
Proposition 4.4.7. Suppose that ũ : H → R × M is Type (II) ˜ J holomorphic,
in the sense of Definition 4.3.3, with finite energy E(ũ) < ∞ and boundary
condition u(∂H) ∈ L. If, furthermore,
u ∗ dλ = 0.
Then, ũ must be constant.
C
In order to prove Theorem 4.4.1, it will be convenient to re-parametrize
ũ : H → R × M by
˜v : R × [−π, π] −→ R × M
z ↦→ ũ(e z ). (4.1)
This strip ˜v then satisfies the boundary condition v(s ± π) ∈ L. Then, we
will also prove a slightly different result, having to do with the behaviour of a
pseudoholomorphic strip with no contact area :
55

Proposition 4.4.8. Suppose that ũ : R × [−π/2, π/2] → R × M is Type (II) ˜ J
holomorphic, in the sense of Definition 4.3.3, with finite energy E(ũ) < ∞ and
boundary condition u(s ± πi) ∈ L. If, furthermore,
u ∗ dλ = 0
and |∇ũ| ≤ C < ∞, then,
C
ũ = (cs + d, γ(ct))
where γ : [0, T ] → M is a homoclinic orbit and c and d are real constants.
We will use then Proposition 4.4.7 to obtain
Proposition 4.4.9. If ˜v is a finite energy strip as in (4.1), then ˜v has bounded
gradient.
Finally, this will allow us to establish Theorem 4.4.1.
Then, in order to prove Theorem 4.4.4, we will need the following result :
Proposition 4.4.10. Suppose ũ : H → R × M is Type (II) pseudoholomorphic,
as in Definition 4.3.3, with the boundary condition at infinity
u(0 + it) → L as t → ±∞
and finite energy. Suppose ˜v(z) = ũ(e z ) is the corresponding strip, as in (4.1).
Then, ˜v(z) has bounded gradient.
This result, in turn, will allow us to conclude Theorem 4.4.4.
The key fact underpinning all of these results is that Proposition 4.4.6 tells
us that if we have uniform gradient bounds on a sequence of curves, we have a
limit (after taking a subsequence, perhaps). We need then to understand what
happens if we don’t have gradient bounds.
An important technical tool in all of these bubbling off proofs is the following
lemma, due to Hofer (in [21]), whose proof is elementary, but whose use greatly
simplifies a certain number of otherwise difficult arguments. A proof may be
found in [4, Chapter 6], or in [33, Lemma 4.6.4].
Hofer’s Lemma. Let (X, d) be a metric space, f : X → R a non-negative,
continuous function. Fix x ∈ X and δ > 0.
Suppose the closed ball B2δ(x) is complete. Then, there exists a ξ ∈ X and a
positive number ɛ ≤ δ so that :
d(x, ξ) < 2δ, sup f ≤ 2f(ξ),
Bɛ(ξ)
ɛf(ξ) ≥ δf(ξ).
56

Proofs of our results
We are now able to build up the proofs of our propositions, culminating in the
proof of Theorem 4.4.4. The proofs themselves are very similar to the standard
theory of pseudoholomorphic curves in symplectizations, though some extra care
is needed near R × L, our singular region.
Proof of Proposition 4.4.7. Since ũ is pseudoholomorphic (in the smooth sense)
away from its singularities Γ(u, L), we have u ∗ dλ is a non-negative multiple of
the volume form away from these singularities. Thus, for u to have no contact
area, there must exist an orbit of the Reeb vector field, γ, so that
ũ = (a(z), γ(f(z)).
In order for the boundary condition on u to be satisfied, we must have γ is a
homoclinic orbit. By Proposition 4.2.6, we have then the existence of a local
diffeomorphism Φ from a neighbourhood of γ to a standard neighbourhood
V ⊂ R3 with the standard contact form. By the symmetry condition on ˜ J
given by Definition 4.3.1, we have that the diffeomorphism Φ pushes the almost
complex structure ˜ J forward to an almost complex structure ˜ M on R×V , which
extends smoothly to all of R×V . We then have that ˜w(z) = (a(z), Φ(u(z))) is a
pseudoholomorphic half-plane into R×V ⊂ R×R 3 with the standard structure.
Furthermore, w(z) satisfies the boundary condition that w(it) ∈ ℓ0 ∪ ℓ1, for all
t ∈ R. Thus, f(it) = const for all t. In order for ˜w(z) to be pseudoholomorphic,
we must have that a + if is holomorphic. We have the boundary condition :
(a+if)(iR) ∈ R, so we may apply the Reflection Principle to obtain an extension
to a holomorphic plane. In the following, we will use the same notation to denote
this extended holomorphic function, a + if.
We wish to show that a must be constant. We thus consider the case it
is non-constant. We now consider two sub-cases. First we assume |∇a| is
bounded. Then, by Liouville’s theorem, a ′ + if ′ is a constant function on C.
Thus, (a + if) is an affine function on C. We write (a + if)(z) = ic1z + c2. We
have ic1it + c2 ∈ R for all t ∈ R, thus, c1, c2 ∈ R. From this, we conclude that
a(s + it) = −c1t + c2. We now check the finite energy condition :
ũ
H
∗
d(φλ) = ˜w
H
∗ d(φαstd)
= φ ′
(a(z)) ( ∂
∂s a)2 + ( ∂
∂t a)2
ds ∧ dt
= 1
2
H
φ
C
′ (−c1t + c2)c 2 1 ds ∧ dt
57

For this to be finite, we need that c1 = 0, which contradicts the assumption
that a was non-constant.
We now consider the case that |∇a| is unbounded. Let Rn > 0 be a monotone
sequence of values with Rn → ∞ and Rn > |∇(a + if)(0)|. Choose then a
sequence ɛn so that ɛn → 0 and ɛnRn → ∞. Then, there exist a sequence
of points zn ∈ C (without loss of generality, we may assume zn ∈ H) so that
|∇(a + if)(zn)| = Rn.
By Hofer’s Lemma, we may move the zn slightly and adjust the Rn and ɛn
so that we we have that there exist zn, ɛn and Rn with the properties :
|∇(a + if)(zn)| = Rn,
Rn → ∞ and ɛn → 0 with ɛnRn → ∞
|∇(a + if)(z)| ≤ 2Rn for |z − zn| ≤ ɛn.
We consider now the sequence of holomorphic functions
Fn := (a + if)(zn + z
Rn
) − (a + if)(zn).
We observe that |∇Fn(0)| = 1 and |∇Fn(z)| ≤ 2 for all |z| ≤ ɛnRn. For each
R > 0, the sequence of functions Fn is bounded in C1 (BR) (once n is large
enough that Fn is defined on BR). We obtain then higher order bounds by
Cauchy’s integral formula. Thus, for each N, a subsequence of Fn converges in
C∞ on BN to a holomorphic function. By a diagonal subsequence argument, we
obtain that the sequence Fn converges in C ∞ loc
to a pseudoholomorphic plane F
with uniform gradient bound and |∇F (0)| = 1. This limiting curve has no more
energy than the original (a + if), so the same argument as in the previous case
gives that F must be constant. This contradicts however that |∇F (0)| = 1.
Proof of Proposition 4.4.8. As in the proof of Proposition 4.4.7, we may write
our pseudoholomorphic curve as :
ũ(z) = (a(z), γ(f(z))),
where γ is a an orbit of the singular Reeb vector field and (a + if) is a holomorphic
function. The boundary condition gives that γ intersects L. We use
Proposition 4.2.6, to obtain a pseudoholomorphic curve ˜w = (a(z), Φ(u(z))).
We have then
˜w : R × [−π/2, π/2] −→ R × V
w(s ± iπ/2) ∈ ℓ0 ∪ ℓ1.
58

We then have that f(s ± iπ/2) is constant on each boundary component. By
assumption, f must have bounded gradient. Furthermore, we have that f is
harmonic, so we have that f(s + it) = c1t + c2, for a pair of real constants c1, c2.
It follows then that a(s + it) = c1s + c ′ 2, where c ′ 2 is another real constant. By
changing the initial condition of γ, we obtain then that
ũ(z) = (c1s + c2, γ(c1t)).
Proof of Proposition 4.4.9. We will prove this result by contradiction. Then,
there exists a sequence of points zn = (sn, tn) so that |∇ũ(zn)| → ∞. Since we
assumed ũ was obtained by re-parametrizing a half-plane, we have sn → +∞.
Furthermore, by Hofer’s lemma, we may assume that we have
|∇ũ(zn)| = Rn, Rn → ∞ and ɛn → 0 with ɛnRn → ∞
|∇un(z)| ≤ 2Rn for z ∈ B(zn, ɛn) ∩ H.
There are now two cases : in case (1), we have interior bubbling, and in case
(2), we have boundary bubbling.
(1) We have that B(zn, ɛn) ⊂ R × (−π/2, π/2), for n large. We introduce a
re-scaled sequence of maps by :
ũn(z) := (a(zn + z
Rn
) − a(zn), u(zn + z
By construction, these maps have uniform gradient bounds on B(0, ɛnRn)
and have |∇un(0)| = 1. We also verify that ũn|B(0,ɛnRn) has energy
bounded above by E(ũ). We then obtain that ũn converge in C ∞ loc to
a Type (II) pseudoholomorphic plane ˜w : C → R × M, in the sense of
Definition 4.3.3. Furthermore, |∇ ˜w(0)| = 1, so ˜w is non-constant.
We have again one of two possibilities : after taking a subsequence, we
may take either B(zn, ɛn) ∩ Γ(u, L) = ∅ for all n, or Γ( ˜w, L) = ∅. Let us
consider these sub-cases separately.
(i) If B(zn, ɛn) ∩ Γ(u, L) = ∅ for all n, the re-scaled curves have image
in the almost complex manifold R × (M \ L). Thus, the theory of
pseudoholomorphic curves into symplectizations, with smooth almost
complex structures, applies. In this case, a finite energy plane with
no dλ contact area must be constant (by [4, Prop. 6.4.2], whose
statement and proof is very similar to our Proposition 4.4.7). This
contradiction eliminates this case.
59
Rn
))

(ii) We have that
C w∗ dλ = 0. Hence w(z) = (b(z), γ(f(z))), where
a + if is holomorphic and γ is an orbit of the Reeb vector field.
By virtue of the fact that Γ(w, L) = ∅, we must have that γ
is a homoclinic orbit. We may then consider the curve ˜v(z) =
(b(z), Φ(γ(f(z)))). This is a pseudoholomorphic plane into the symplectization
R × V , holomorphic with respect to a smooth almost
complex structure. Then, again by [4, Prop. 6.4.2], it follows that
˜v must be constant. Hence, ˜w is constant. This contradiction eliminates
this case, and thus the possibility of interior bubbling. We are
now only left with the case of boundary bubbling.
(2) After taking a subsequence, we may assume that zn = sn + itn have tn →
±π/2. Let us assume tn → π/2. (The other case is similar.) Furthermore
(by taking a subsequence), we may assume that |π/2 − tn|Rn → c < ∞.
Otherwise, we could make ɛn < |π/2 − tn| and we would be in the case of
interior bubbling.
Let ζn = sn + iπ/2. Let δn = ɛn − |π/2 − tn|. Then, we have δnRn → ∞
and B(ζn, δn) ⊂ B(zn, ɛn). We now introduce a sequence of curves, ũn as
follows :
ũn : H ∩ B(0, ɛnRn) −→ R × M
z ↦→ (a(ζn + −iz
) − a(ζn), u(ζn + −iz
)).
Rn
By construction, these curves satisfy a uniform gradient bound. Thus,
we obtain that ũn converge in C∞ loc to a Type (II) pseudoholomorphic
curve ˜w in the sense of Definition 4.3.3. We have that ˜w : H → R × M
and ˜w maps ∂H into R × L, with w∗ dλ = 0. Furthermore, we have
|∇ũn(0+iRn(tn −π/2))| = 1. Since |tn −π/2|Rn → c < ∞, by considering
a subsequence, we have that Rn(tn − π/2) → z∞, with |∇ ˜w(z∞)| = 1.
Thus ˜w is non-constant. This contradicts Proposition 4.4.7. This last
contradiction completes the proof.
Proof of Theorem 4.4.1. Let ˜v = (b, v) be the pseudoholomorphic strip obtained
by the change of variables (4.1). Let Sk = log Rk. We will show that after con-
T t
sidering a subsequence, v(Sk +t) is asymptotic to γ( ), where γ is a homoclinic
2π
orbit of the Reeb vector field.
By Proposition 4.4.9, we have that there exists a constant so that |∇v(z)| ≤
C < ∞ for all z ∈ R × [−π/2, π/2].
60
Rn

We consider now the sequence of curves :
˜vn : R × [−π/2, π/2] −→ R × M
z ↦→ (b(Sk + z) − b(Sk), v(Sk + z)).
We have then that |∇˜vn| ≤ C < ∞. Thus, there exists a Type (II) pseudoholomorphic
curve in the sense of Definition 4.3.3 ˜w, so that ˜vn → ˜w in C∞ loc . We
observe that
w
[−R,R]×[−π/2,π/2]
∗ dλ = lim
n→∞
= lim
n→∞
= 0
vn
[−R,R]×[−π/2,π/2]
∗ dλ
[Sn−R,Sn+R]×[−π/2,π/2]
v ∗ dλ
and thus we obtain a finite energy strip with no contact area and bounded
gradient. It then follows from Proposition 4.4.8 that ˜w = (cs + d, γ(ct)), where
γ is a homoclinic orbit and c and d are constants.
From this, we now compute that for any φ : R → [0, 1] with φ ′ ≥ 0 and
φ(−∞) = 0 and φ(+∞) = 1, we have :
T =
˜w
R×[−π/2,π/2]
∗ d(φλ)
=
φ ′ (cs + d)c 2 ds ∧ dt
= cπ
R×[−π/2,π/2]
Hence, we have that w(Sk + it) → γ(T t/π) as Sk → ∞, and the convergence is
in C ∞ ([−π/2, π/2], M).
Proof of Proposition 4.4.10. We will prove this result by contradiction. Then
there are a sequence of points zn = (sn, tn) so that |∇˜v(zn)| = Rn → ∞. We
observe that the argument in the proof of Proposition 4.4.9 that establishes the
contradiction from interior bubbling applies in this case as well. Thus, after
taking a subsequence, we may assume that zn = sn + itn have tn → ±π/2. Let
us assume tn → π/2. (The other case is similar.) Furthermore, we may assume
that |π/2 − tn|Rn → c < ∞. Let ζn = sn + iπ/2. Let δn = ɛn − |π/2 − tn|. Then,
we have δnRn → ∞ and B(ζn, δn) ⊂ B(zn, ɛn). We now introduce a sequence of
curves, ũn as follows :
ũn : H ∩ B(0, ɛnRn) → R × M
z ↦→ (a(ζn + −iz
) − a(ζn), u(ζn + −iz
)).
61
Rn
Rn

By construction, these curves satisfy a uniform gradient bound. Thus, we
obtain that ũn converge in C∞ loc to a Type (II) pseudoholomorphic curve ˜w in
the sense of Definition 4.3.3. By re-parametrizing the domain, we have that
˜w : H → R × M and ˜w maps ∂H into R × L, with w∗ dλ = 0.
We have
w(0 + it) = lim un(0 + it) = lim u(ζn + t
= lim u(sn + t
Rn
+ i π
2 )
By the boundary condition on u, we have that there is a subsequence along
which u(sn + t ) converges to a point on L. Thus, w(0 + it) ∈ L for each t ∈ R.
Rn
We then have that ˜w satisfies the hypotheses of Proposition 4.4.7. However,
˜w is non-constant, by construction.
Proof of Theorem 4.4.4. Let ˜v = (b, v) be the pseudoholomorphic strip obtained
by the change of variables (4.1). Let Sk = log Rk. We will show that after con-
T t
sidering a subsequence, v(Sk +t) is asymptotic to γ( ), where γ is a homoclinic
2π
orbit of the Reeb vector field.
By Proposition 4.4.10, we have that there exists a constant so that |∇v(z)| ≤
C < ∞ for all z ∈ R × [−π/2, π/2].
We consider now the sequence of curves :
˜vn : R × [−π/2, π/2] → R × M
Rn
z ↦→ (b(Sk + z) − b(Sk), v(Sk + z)).
We have then that |∇˜vn| ≤ C < ∞. Thus, there exists a Type (II) pseudoholomorphic
curve in the sense of Definition 4.3.3 ˜w, so that ˜vn → ˜w in C∞ loc . We
observe that
w
[−R,R]×[−π/2,π/2]
∗
dλ = lim
vn
n→∞
[−R,R]×[−π/2,π/2]
∗ dλ
= lim
v
n→∞
∗ dλ
= 0
[Sn−R,Sn+R]×[−π/2,π/2]
and thus we obtain a finite energy strip with no contact area and bounded
gradient.
62
)

Furthermore, we have that this strip ˜w has boundary in R × L. Indeed, for
any s, we have :
w(s ± iπ/2) = lim
n→∞ vn(s ± iπ/2)
= lim
n→∞ v(Sn + s ± iπ/2)
= p ∈ L by the asymptotic boundary condition on v.
It then follows from Proposition 4.4.8 that ˜w = (cs + d, γ(ct)), where γ is a
homoclinic orbit and c and d are constants.
From this, we now compute that for any φ : R → [0, 1] with φ ′ ≥ 0 and
φ(−∞) = 0 and φ(+∞) = 1, we have :
T =
˜w
R×[−π/2,π/2]
∗ d(φλ)
=
φ ′ (cs + d)c 2 ds ∧ dt
= cπ
R×[−π/2,π/2]
Hence, we have that w(Sk + it) → γ(T t/π) as Sk → ∞, and the convergence is
in C ∞ ([−π/2, π/2], M).
4.4.2 Exponential rate of convergence
Let us assume we have a pseudoholomorphic half-plane, ũ : H → R × M,
satisfying the hypotheses of Theorem 4.4.1. Furthermore, let us assume that
for some sequence Rk → ∞, u(Rke it ) converges to a non-degenerate homoclinic
orbit. Then, we may conclude that u(re it ) has a limit as r → ∞. Furthermore,
we have that u(re it ) approaches the asymptotic limit exponentially fast.
Finally, we will show that the component of u transversal to its asymptotic
limit will approach an eigenvector of a suitable unbounded operator, at an
exponential rate governed by the corresponding eigenvalue. These results are
similar in flavour to results in [2], [24] and [35].
This result is in keeping with the Floer theoretic analogy between pseudoholomorphic
curves and gradient flow lines. The underlying idea is that the
pseudoholomorphic curve equation is the gradient flow equation for an action
functional on the loop space. This action functional is not Morse-Smale, and
so the gradient flow is not well posed. However, we may formally write the
gradient flow as the non-linear Cauchy-Riemann equations. The analogy proves
63

accurate in many cases — this exponential rate of convergence being one of
them.
The first result is that a pseudoholomorphic half-plane, if it is asymptotic
as in Theorem 4.4.1 to a non-degenerate homoclinic orbit, then the asymptotic
limit is the same no matter what sequence of Rk we use.
Theorem 4.4.11. Let M, λ and ˜ J as in Definition 4.3.1. Suppose that ũ :
H → R × M is Type (II) ˜ J holomorphic, in the sense of Definition 4.3.3, with
finite energy E(ũ) = T < ∞ and boundary condition u(∂H) ∈ L. Suppose that
γ : [0, T ] → M is a non-degenerate homoclinic orbit so that for some sequence
Rk → ∞
u(Rke it ) → x(T t/π).
Then, for any sequence R ′ k
→ ∞,
u(R ′ ke it ) → x(T t/π).
Proof. Let ℓu and ℓs be unit unstable and stable Legendrians of action δ, for
some δ > 0, small, as given by Lemma 3.2.2. Then, we have x(δ) ∈ ℓu and x(T −
δ) ∈ ℓs. We recall that one of the equivalent formulations of non-degeneracy
for a homoclinic orbit is that the image of [0, T − δ] × ℓu → M : (t, p) ↦→ φt(p)
intersect ℓs transversely at x(T − δ). It follows then that the homoclinic orbit x
is isolated among homoclinic orbits of nearby action. Let U be a neighbourhood
of x in the space of paths in M, with respect to the C ∞ topology so that there
are no other homoclinics of nearby action in U. Let V ⊂ U be a smaller
neighbourhood of x.
We will proceed by contradiction. Suppose that u(R ′ k eit ) does not converge
to x(T t/π). By taking a subsequence, we may assume that u(R ′ k eit is bounded
away from x(T t/π) in the C ∞ topology on paths in M. Then, by Theorem
4.4.1, R ′ k
has a subsequence R′′
k
so that u(R′′
k eit ) converges to a y(T t/π). We
must have y = x and thus y /∈ U. We observe that the map S ↦→ u(Se it ) is
a continuous map in the C ∞ topology on paths in M. Thus, we must have
infinitely many S so that u(Se it ) /∈ V but u(Se it ) ∈ U. By Theorem 4.4.1,
we then have a subsequence Sk along which u(Ske it ) → z(T t/π), where z is a
homoclinic orbit. As z /∈ V , z = x. But there are no homoclinic orbits of action
T in U. The result now follows from this contradiction.
Theorem 4.4.12. Let M, λ and ˜ J as in Definition 4.3.1. Let ũ be a pseudoholomorphic
strip, as in Theorem 4.4.11, and γ be its asymptotic limit. Suppose
ı : Dr ↩→ M \ L is an embedded disk, transverse to the Reeb vector field, so that
ı(0) is a point on γ, and so that ı ∗ dλ = dx ∧ dy.
64

Let x(s, t) and y(s, t) be the projections of u(s, t) on to Dr by means of the
Reeb flow, where these are defined. Let z(s, t) so that φz(s,t)(ı(x(s, t), y(s, t)) =
u(s, t). (We note this is not a priori defined for all s, t.)
Then, there exist constants a0 ∈ R, r > 0, so that for each k, l ∈ N, and
for each 0 < ρ < min{r/2, 1/T }, there exist constants ck,l so that for all ɛ > 0,
there exists Sɛ so that for all s ≥ Sɛ, x(s, t), y(s, t) and z(s, t) are defined for
s ≥ Sɛ and we have
sup |∂
0≤t≤T
k s ∂ l tx(s, t)| ≤ ck,le −rs
sup |∂
0≤t≤T
k s ∂ l ty(s, t)| ≤ ck,le −rs
sup |∂
0≤t≤T
k s ∂ l t(z(s, t) − t)| ≤ ck,le −ρs
sup |∂
0≤t≤T
k s ∂ l t(a(s, t) − a0 − s)| ≤ ck,le −ρs .
We may say a bit more, in fact : we may find a representation of ζ(s, t) :=
(x(s, t), y(s, t)) in terms of an eigenvector of an unbounded operator. To make
this precise, we will put very specific symplectic coordinates on the disk, Dr.
Let ℓu and ℓs be the unit unstable and stable Legendrians, respectively. Then,
projecting them on Dr by means of the Reeb flow, we have intersecting Lagrangians.
Since γ was assumed non-degenerate, they intersect transversally.
We may make a symplectic change of variables so they are given by R and
iR respectively. We let (x, y) denote the corresponding coordinates. We let
ζ(s, t) = (x(s, t), y(s, t)), the projection of u(s, t) to the disk by following the
Reeb flow.
We now introduce a family of almost complex structures on Dr by using
the projection along the Reeb flow : for each z, we define M(x, y, z) to be
the almost complex structure on Dr given by projecting the almost complex
structure ˜ J|ξ using the Reeb flow to time z. For each z, M(·, ·, z) is an almost
complex structure on Dr, compatible with dx ∧ dy.
We have then that ζ satisfies the following non-linear Cauchy–Riemann equation
:
∂
∂
ζ + M(ζ, z(s, t)) ζ = 0.
∂s ∂t
Let M0(t) = M(0, 0, t). We introduce the function space
W 1,2
ℓ ([0, T ], R 2 ) = {f ∈ W 1,2 ([0, T ], R 2 ) | f(0) ∈ R · e1 and f(1) ∈ R · e2}
and we equip L 2 ([0, T ], R 2 ) with the inner product T
0 ( dx ∧ dy)(·, M0(t)·) dt.
(This gives a norm equivalent to the standard norm.)
65

We then define an unbounded linear operator by :
We now have :
A∞ : L 2 ([0, T ], R 2 ) ⊃ W 1,2
ℓ ([0, T ], R 2 ) −→ L 2 ([0, T ], R 2 )
γ ↦→ −M0(·) ˙γ(·).
Theorem 4.4.13. If ζ is not identically zero,
where
R s
s Λ(τ) dτ
ζ(s, t) = e 0 (e(t) + r(s, t))
A∞ is self-adjoint (with respect to the new inner product),
e ∈ W 1,2
ℓ ([0, T ], R 2 ) is an eigenvector of A∞ corresponding to an eigenvalue
λ < 0,
Λ(s) : [s0, ∞) → R is a smooth function with Λ(s) → λ as s → ∞,
r : [s0, ∞) × [0, T ] → R decays to zero uniformly in t as s → ∞, together with
all of its derivatives.
In order to prove the asymptotic convergence results, we will use work done
by Abbas in [2]. In [2], he considers a (smooth) three dimensional contact
manifold with contact form λ and a Legendrian knot ℓ. He then considers the
boundary value problem of considering a pseudoholomorphic half-plane ũ : H →
R×M with Lagrangian boundary condition u(0+it) ∈ ℓ. He shows that a finite
energy half-plane of this kind is asymptotic to a Reeb chord. Furthermore, if the
Reeb chord is non-degenerate (analogous to our condition in Definition 4.4.2),
he proves versions of our theorems 4.4.12 and 4.4.13. Since these analytical
questions are local in nature, we may use our local model of the neighbourhood
of a homoclinic orbit, Proposition 4.2.6, in order to reduce our problem to the
problem studied by Abbas.
We cite the following results from Abbas, in [2]. The original statements of
these theorems are in a more general setting, but his proofs reduce the problem
to the local situation described here.
In the following two theorems, we consider a neighbourhood, V , in R 3 of
{(0, 0)} × [0, T ], for some T > 0. We have two Legendrian curves in V , ℓ0
and ℓ1, with ℓ0 = R × {(0, 0)} ∩ V and with ℓ1 = {(x(τ), τ, z(τ) | τ small}
and x(0) = 0 and z(0) = T . This is the local model of any non-degenerate
Legendrian chord, in the sense of [2]. In our case, this is the image of the local
diffeomorphism given by Proposition 4.2.6.
66

Theorem 4.4.14 ([2, Theorem 1.3]). Suppose ũ : [s0, ∞) × [0, T ] → R × V
is pseudoholomorphic with respect to a smooth almost complex structure ˜ J, adjusted
to λ, with finite energy, and satisfies u(S + it) → (0, 0, t) as S → ∞. Let
us write u(s, t) = (x(s, t), y(s, t), z(s, t)). Then, there exist constants a0 ∈ R,
r > 0, S ≥ s0 so that for each k, l ∈ N, and for each 0 < ρ < min{r/2, 1/T },
there exist constants ck,l so that for all s ≥ S, we have :
sup |∂
0≤t≤T
k s ∂ l tx(s, t)| ≤ ck,le −rs
sup |∂
0≤t≤T
k s ∂ l ty(s, t)| ≤ ck,le −rs
sup |∂
0≤t≤T
k s ∂ l t(z(s, t) − t)| ≤ ck,le −ρs
sup |∂
0≤t≤T
k s ∂ l t(a(s, t) − a0 − s)| ≤ ck,le −ρs
This result is extended to a representation formula for the component of u
transverse to the Reeb vector field. First, we consider the symplectic surface
Σ := {z = 0}. This is transverse to the Reeb vector field, so we may use
the Reeb flow to map any point (x, y, z) to (x, y, 0). Furthermore, by projecting
along ∂
∂z , we may find an isomorphism between ξ(x,y,0 and T(x,y,0)Σ. This induces
a symplectic form on the surface, which in this case is dx∧ dy. Furthermore, We
see that ℓ0 and ℓ1 are Lagrangian curves in Σ. Thus, we may find a symplectic
change of variables so that ℓ0 = R · ∂
∂x ′ and ℓ1 = R · ∂
∂y ′ . For p ∈ V , let ρp
denote the induced map from ξp to R 2 . Then, we introduce the almost complex
structure M(x, y, z) by ρx,y,z ◦ J(x, y, z) = M(x, y, z) ◦ ρp. For each fixed z,
this is an almost complex structure on Σ compatible with the symplectic form
dx ∧ dy.
Let ζ(s, t) be the representation of (x(s, t), y(s, t)) in the x ′ , y ′ coordinates
introduced on Σ. We have then that ζ satisfies the following non-linear Cauchy–
Riemann equation :
∂
∂
ζ + M(ζ, z(s, t)) ζ = 0.
∂s ∂t
Let M0(t) = M(0, 0, t). We introduce the function space
W 1,2
ℓ ([0, T ], R 2 ) = {f ∈ W 1,2 ([0, T ], R 2 ) | f(0) ∈ R · e1 and f(1) ∈ R · e2}
and we equip L 2 ([0, T ], R 2 ) with the inner product T
0 ( dx ∧ dy)(·, M0(t)·) dt.
(This gives a norm equivalent to the standard norm.)
We then define an unbounded linear operator by :
A∞ : L 2 ([0, T ], R 2 ) ⊃ W 1,2
ℓ ([0, T ], R 2 ) −→ L 2 ([0, T ], R 2 )
γ ↦→ −M0(·) ˙γ(·).
67

Then, the result of Abbas gives :
Theorem 4.4.15 ([2, Theorem 1.4]). If ζ is not identically zero,
where
ζ(s, t) = e
R s
s Λ(τ) dτ
0 (e(t) + r(s, t))
A∞ is self-adjoint (with respect to the new inner product),
e ∈ W 1,2
ℓ ([0, T ], R 2 ) is an eigenvector of A∞ corresponding to an eigenvalue
λ < 0,
Λ(s) : [s0, ∞) → R is a smooth function with Λ(s) → λ as s → ∞,
r : [s0, ∞) × [0, T ] → R decays to zero uniformly in t as s → ∞, together with
all of its derivatives.
Remark 4.4.16. Robbin and Salamon, in [35], provide a summary of the key
ideas of the proofs of these results, which are carried out carefully in [2].
Now, finally, we are able to prove our versions of these theorems. Our proofs
are straightforward combinations of Abbas’s work with our local model theorem.
Proof of Theorems 4.4.12 and 4.4.13. Denote by Φ the local diffeomorphism
given by Proposition 4.2.6, and U, the neighbourhood of γ on which it is defined,
and V ⊂ R 3 , its image. We denote by ℓ0 and ℓ1, the images of L at γ(0)
and γ(T ). We may take Φ so that Φ is a diffeomorphism in a neighbourhood of
the image of Dr.
By Theorem 4.4.11, u(s, t) has image in U for s sufficiently large. Let
˜v = (a, Φ(u). We have that ˜v is a pseudoholomorphic curve into R × V, with
boundary in R × ℓi, i = 0, 1. We write v(s, t) = (x ′ (s, t), y ′ (s, t), z ′ (s, t)) where
(x ′ , y ′ ) are the coordinates as in the result of Abbas (see Theorem 4.4.14).
We observe that where the coordinates x, y and z are defined (as in the
theorem statement), Φ(x, y, z) = (x ′ , y ′ , z ′ ).
We may now apply the result of Abbas, Theorem 4.4.14. By virtue of
the exponential convergence of z(s, t) to t, we have that for each t in the
interior of (0, T ), for s large enough, z ′ (s, t) > 0. It then follows that
(x ′ (s, t), y ′ (s, t), z ′ (s, t)) have a unique lift, and Theorem 4.4.12 now follows.
Similarly, we apply Theorem 4.4.15 to ˜v. Again, we have that for each
compact subset of (0, T ), for s sufficiently large, there is a unique lift, and thus
Φ tells us the whole story.
Remark 4.4.17. We note that in this proof, we have a good estimate on how
large s must be. An avenue for future work is to study how the strip winds
about the singular Legendrian. A good control on this will allow for a stronger
asymptotic statement.
68

4.5 Future work
This work with singular almost complex structures raises many questions. The
first question is about the relationship between Type (I) and Type (II) pseudoholomorphic
curves. This could potentially lead to a new generalization of the
Gromov removal of singularities theorem. Another problem that needs addressing
is the compactification of the space of Type (II) pseudoholomorphic curves.
And fundamental is the question of the relationship between these curves and
the (standard) pseudoholomorphic curves on nearby regular energy levels.
All of these questions come out of the larger background project to understand
the change in contact homology caused by Legendrian surgery. Ultimately,
the goal is to understand the relationship between the various relevant symplectic
and contact invariants : contact homology for M ± , symplectic field theory
for the symplectic cobordism given by W , and the yet-to-be-developed homoclinic
contact homology. In turn, this should give a lot of information about
Hamiltonian dynamics in relation to these invariants.
69

Chapter 5
Energy quantization for
pseudoholomorphic curves with
a relaxed area bound
5.1 Overview
We are interested in studying the behaviour of a pseudoholomorphic plane
ũ : C → R × M
with merely a bound on the contact area :
0 < u ∗ dλ = C < ∞. (5.1)
We note that such a plane is not, a priori, a finite energy curve. We recall that
a pseudoholomorphic curve ũ is a finite energy curve if
sup{ ũ ∗ d(φλ) | φ : R → [0, 1] with φ ′ ≥ 0} < ∞.
We denote the left hand side of this inequality by E(ũ), and call it the E-energy
of the curve ũ. This energy was introduced by Hofer in [21]. Finiteness of this
energy is necessary for all of the existing work on compactness for pseudoholomorphic
curves in symplectizations. Infinite energy curves can be very badly
behaved (as our example below will show) — in particular, such a map will not
be proper!
We will prove the following result :
70

Theorem 5.1.1. Suppose ũ : C → R × M is pseudoholomorphic with respect to
an almost complex structure ˜ J, adjusted to the contact form λ, and has finite
contact area :
0 < u ∗ dλ = C < ∞.
C
We also impose the technical condition that πλ T u never vanishes.
Then, there exists a periodic orbit of the Reeb vector field with action 0 < T ≤ C.
This result establishes an energy threshold for the contact area of a pseudoholomorphic
curve. This paves the way for establishing an energy quantization
result. This is part of an ongoing project with Abbas and Hofer to understand
the behaviour of these curves with infinite E-energy, but finite dλ contact area.
The theory of pseudoholomorphic curves into symplectizations with an Eenergy
bound is very well developed, with a good understanding of the compactification
of the space of such curves [6] and of how they may be perturbed
[25]. The study of them has led to some very interesting results in dynamical
systems, for instance [27], and to the proof of the Weinstein conjecture for many
classes of contact manifolds.
In order to solve the Weinstein conjecture for all contact 3-manifolds, Hofer
has introduced a programme to study a generalization of the pseudoholomorphic
curve equation. The goal is to exploit a result of Giroux’s in order to obtain
existence for a finite energy curve. Giroux proved that every contact structure
on a 3–manifold is supported by an open book [18]. Furthermore, Giroux’s construction
exhibits a “nice” contact form generating the contact structure. The
idea, then, is to take such an open book, with its contact form, and construct
a pseudoholomorphic curve by a construction on a leaf of the open book. This
exploits the special structure of the Giroux contact form. This construction is
done by Abbas, in [1]. One then takes a homotopy from the Giroux contact form
to the contact form of interest. The goal is to show that a pseudoholomorphic
curve persists through the homotopy.
The case in which the leaves of the Giroux foliation have no genus is taken
care of in [3]. The case in which the leaves have genus is more subtle. The
Cauchy–Riemann operator is Fredholm, but with negative index. Thus, there
is no way to obtain transversality. Instead, as discussed in [3], Hofer has introduced
a twisted pseudoholomorphic curve equation.
We let (Σ, j) be a Riemann surface, and J an almost complex structure on
the contact structure, compatible with the symplectic form dλ|ξ. We have a
finite set of punctures Γ ⊂ Σ and denote ˙ Σ := Σ\Γ. Let πλ be the projection to
ξ = ker λ along the Reeb vector field. The goal is to show existence of a tuple
71

(Σ, j, Γ, ũ, γ), where
ũ = (a, u) : ˙ Σ −→ R × M
πλ T u ◦ j = J ◦ πλ T u on ˙ Σ
(u ∗ λ) ◦ j = da + γ on ˙ Σ
and dγ = 0 = d(γ ◦ j) on Σ.
This PDE reduces to the standard Cauchy–Riemann equation if γ = 0. The
energy used for this generalized Cauchy–Riemann equation remains the same
as before :
E(ũ) = sup{
ũ ∗ d(φλ) | φ : R → [0, 1] and φ ′ ≥ 0}.
The key point being that this energy does not keep track of the harmonic form
γ.
This new problem is Fredholm, of the correct index and can be made surjective.
However, the existing compactness theory is no longer applicable. In
particular, one of the objects which may bubble off during the homotopy is a
pseudoholomorphic plane with no bound on the E-energy. The contact area,
however, will be non-zero, and bounded. This is the problem that leads us to
study planes with control only on the contact area (5.1).
Remark 5.1.2. There exist pseudoholomorphic curves with (non-zero) finite contact
area (5.1), but infinite E-energy. Let (M, ξ) be a contact manifold with
contact form λ. Suppose there exists an adjusted almost complex structure ˜ J
on R×M, invariant under the Reeb flow (this is a strong assumption). Suppose
also that there exists a finite energy plane ũ0 = (a0, u0) : C → R × M. We now
observe that if f + ig : C → C is holomorphic (classically) then
˜v(z) := (a0(z) + f(z), φf(z) ◦ u0(z))
is also pseudoholomorphic.
We may take f + ig so that |∇f| is bounded from below, for |z| ≥ R (for
instance, f + ig = zk ). We observe that
E(˜v) ≥ sup{ φ ′ (b(s, t))|∇b(s, t)| 2 ds dt | φ : R → [0, 1] and φ ′ ≥ 0}.
C
We have |∇b(s, t)| 2 ≥ |∇f(s, t)| 2 ≥ c > 0 for |z| ≥ R. Let us also restrict
our attention to φ for which φ ′ has compact support. Then, we have (where µ
72

denotes the Lebesgue measure on R2 ):
φ ′ (b(s, t))|∇b(s, t)| 2
ds dt ≥ c φ ′ (b(s, t)) ds dt
C
= c
= c
|z|≥R
|z|≥R
b(s,t)
−∞
φ ′′ (τ) dτ ds dt
φ
R
′′ (τ)µ{(s, t) | b(s, t) ≥ τ and s 2 + t 2 ≥ R 2 } dτ.
From the asymptotic formula for a(s, t), as derived in [24], a(z) is asymptotic
to c1 log(|z| 2 )+c2, where c1 > 0. Thus, {(s, t) | b(s, t) ≥ τ} has infinite measure.
We may choose a φ for which φ ′′ is positive on an interval. It follows that the
E energy of v is now infinite. It also follows by Stokes theorem that the contact
area of ˜v is the same as the contact area of ũ. Thus, ˜v has infinite E-energy,
but finite contact area.
An example of a manifold and contact form admitting an almost complex
structure invariant under the Reeb flow is S 3 with the standard contact form
(corresponding to the Hopf fibration). The symplectization R × S 3 may be
identified with C 2 \ {0}. The standard complex structure on C 2 is adjusted to
the contact form. This complex structure is invariant under the action of the
Reeb flow.
5.2 “Renormalization” by the Reeb flow
The proof of Theorem 5.1.1 uses a key construction, which we call renormalization.
It is inspired by the example in Remark 5.1.2, where the infinite energy
pseudoholomorphic curve is a graph over a finite energy curve, by means of the
Reeb flow.
We let
0 < C = u ∗ dλ < ∞.
C
We will consider our original infinite E-energy curve, ũ, restricted to a disk
of radius R. We now consider the immersed cylinder L obtained by flowing the
image of the boundary, u(∂DR), by means of the Reeb flow. Our goal is to
find a pseudoholomorphic disk ˜v = (b, v) for which b(∂DR) = const and with
boundary in L (of degree 1 on the S1 factor). In order to construct this disk, we
will consider a homotopy through pseudoholomorphic curves with boundary in
the cylinder L. By Stokes theorem, we have that the dλ area of any curve with
(degree 1) boundary in L is the same as the dλ area of ũ|DR . Furthermore, by
73

the boundary behaviour of ˜v and Stokes theorem, we have that
E(˜v) = v ∗
dλ = u ∗ dλ ≤ C < ∞.
Thus, we have an a priori bound on the E-energy of the curve ˜v obtained in
this way. This new curve, ˜v, represents a renormalization of ũ|DR by means of
the Reeb flow.
In order to prove Theorem 5.1.1, we describe a boundary value problem
corresponding to finding a homotopy from our original curve ũ|DR to the renormalized
˜v :
⎧
⎪⎨
⎪⎩
(b, v) : DR → R × M
πλ T v ◦ i = J(v) ◦ πλ T v
v ∗ λ ◦ i = db
b(z) = τ a(β(z)) for z ∈ ∂DR
where β : ∂DR → ∂DR is of degree 1
v(z) = φf(z)(u(β(z))) for z ∈ ∂DR
where f : ∂DR → R, and f(R) = 0.
DR
(5.2)
Our original curve corresponds to the homotopy parameter τ = 1. The
renormalized curve corresponds to τ = 0.
We prove that if we have a solution for a value τ0 > 0, then there exist
solutions for τ sufficiently close to τ0. This follows by a straightforward application
of well-known properties of the non-linear Cauchy-Riemann operator. We
discuss this in section 5.3.3.
We also prove that non-compactness of the solution space gives us one of
two outcomes : either there exists a periodic Reeb orbit of period T ≤ C, or
we “short-circuit” the homotopy and there exists a curve ˜v : DR → R × M,
satisfying (5.2) for τ = 0. This is proved in section 5.3.2.
Thus, we reduce the problem to the case that for any R > 0, we may find a
curve ˜vR that renormalizes ũ|DR .
We now take a sequence of Rn → ∞. We consider then our original curve
restricted to the disks of radius Rn, ũn := ũ|Rn. This is now a sequence of
pseudoholomorphic disks with a uniform bound on the contact area, but no
bound on the E energy.
By means of the renormalization, for each Rn, we obtain a curve ˜vn. These
curves all have uniformly bounded E-energy. Thus, we either have gradient
bounds, and thus they converge to a pseudoholomorphic plane with finite Eenergy,
or they bubble off a pseudoholomorphic plane with finite E-energy.
Then, by the usual compactness results for finite energy pseudoholomorphic
curves, there must exist a Reeb orbit of period T ≤ C.
74

5.3 Proof of the result
In the following, M will denote a three dimensional manifold equipped with a
contact form λ, generating the contact structure ξ = ker λ. We will denote by
πλ : T M → ker λ the projection along the Reeb vector field Xλ. Let J : ker λ →
ker λ be a complex structure compatible with dλ. We then define ˜ J to be the
almost complex structure on R × M with ˜ J ∂
∂a = Xλ and ˜ J|ξ = J. We will now
prove
Theorem 5.1.1. Assume we have a non constant solution ũ = (a, u) of
⎧
⎪⎨
⎪⎩
ũ : C −→ R × M
πλ T u ◦ i = J(u) ◦ πλ T u
u ∗ λ ◦ i = da
0 <
C u∗ dλ < ∞
We also assume that πλ T u never vanishes.
Then there exists a periodic trajectory x of the Reeb vector field such that
u ∗
dλ ≥ x ∗ λ.
C
S 1
(5.3)
Our proof will be by contradiction. We assume that no such periodic orbit
exists. We consider then ũ restricted to a disk of radius R. As discussed above,
we will find a homotopy between this and a pseudoholomorphic ˜v = (b, v) on
the disk, with, at the boundary, v = φf ◦ u (where φt denotes the Reeb flow)
and b = constant. The map ˜v will then have both dλ and E energy equal to
the dλ-energy of u|DR .
In this and in the following, we denote DR = {z ∈ C | |z| ≤ R}, the closed
disk of radius R. We will also write D for the closed disk of radius 1.
After repeating this construction for a sequence of R → ∞, we obtain a
sequence of pseudoholomorphic curves with uniformly bounded energy. A subsequence
then converges to a finite energy plane asymptotic to a periodic orbit
of action no greater than
C u∗ dλ. This contradiction then establishes the result.
First, we will describe the boundary value problem that will give us the
homotopy. Then, we will prove a compactness result for the solutions of this
boundary value problem. In particular, we will show that non-compactness
implies the existence of a curve ˜v, which was the “goal” of the homotopy. Finally,
we will prove that this problem is elliptic and satisfies a Fredholm theory.
5.3.1 The boundary value problem
75

We recall that we are interested in studying solutions (˜v, τ) to the boundary
value problem (5.2) :
⎧
⎪⎨
⎪⎩
(b, v) : DR → R × M
πλ T v ◦ i = J(v) ◦ πλ T v
v ∗ λ ◦ i = db
b(z) = τ a(β(z)) for z ∈ ∂DR
where β : ∂DR → ∂DR is of degree 1
v(z) = φf(z)(u(β(z))) for z ∈ ∂DR
where f : ∂DR → R, and f(R) = 0.
(5.2)
We note that for each τ, the boundary condition is totally real. We denote
L := {φt(u(z)) | z ∈ ∂DR , t ∈ R}.
We observe that {0} × L is an immersed Lagrangian submanifold of (R ×
M, d(etλ)). For each τ, we will take the boundary condition to lie in the immersed,
totally real submanifold Lτ := {(τa(z), φt(u(z)) | z ∈ ∂DR , t ∈ R}.
This immersed submanifold may be made Lagrangian by twisting the symplectic
structure on R × M. In this sense, we have a homotopy of Lagrangian
boundary conditions.
We also observe that for τ = 1, ˜v = ũ|DR
is a solution.
We will prove that assuming no periodic orbit exists, for each R > 0, there
must exist a disk ˜v that satisfies (5.2) for τ = 0.
In order to do this, we will first show that if we have a sequence of solutions
(˜vτ, τ), for τ ∈ (0, 1], then either we have a subsequence that converges to a
solution to (5.2), or we “short-circuit” the homotopy and bubble off a curve ˜v
that solves (5.2) for τ = 0. This part involves some careful a priori estimates
and the introduction of a variant on Hofer’s E energy.
We will then show that the set of τ for which a solution exists, is open.
We will suppose we have a solution for some τ0 and show a solution exists for
nearby τ. In order to do this, we will view solutions to the PDE problem (5.2)
as zeros of a section of a Banach space bundle. We will linearize the section at
our solution, and then show that the linearized operator is a surjective Fredholm
operator. This part will be developed in Section 5.3.3. This uses well established
methods, as in for instance [33, 4].
Finally, we will conclude that the existence, for each R, of a solution to (5.2)
for τ = 0 gives us Theorem 5.1.1.
5.3.2 Compactness of disks
76

We will now study compactness of solutions to the problem (5.2). This boundary
value problem features some important non-compactness phenomena. We recall
that pseudoholomorphic curves satisfy an interior regularity estimate so that (in-
terior) gradient bounds gives (interior) C ∞ loc
bounds. Furthermore, they satisfy
a form of Gromov compactness : essentially, non-compactness (gradient blow
up) can be explained by the non-compactness of the space of bi-holomorphic
maps on the domain.
The most difficult non-compactness problem comes from the boundary condition.
We require our curves to have boundary in an immersed Lagrangian
whose image in the manifold can be quite wild. In order to deal with this, we
will establish an a priori bound on how far the boundary condition can flow.
In other words, we will show that the solutions to (5.2) only “see” a finite piece
of the boundary condition L, an immersed [−T, T ] × S1 for some T > 0, large.
The easiest problem to deal with is interior bubbling. In this case, we conclude
that interior bubbling gives the existence of a finite energy plane. This,
by [21], is then asymptotic to a periodic orbit whose action equals the dλ area
of the plane. If this were to occur, we would then be done.
Finally, we deal with the problem of boundary bubbling. In this case, by
virtue of the a priori bound that forces our curves to have boundary in a
finite portion of L, we have that if we have boundary bubbling, after rescaling,
we obtain a curve that satisfies (5.2) for τ = 0. In a sense then, boundary
bubbling immediately “short-circuits” the renormalization process and gives us
the renormalized curve without going through the whole homotopy. In this
case, we don’t obtain compactness. Instead, we have that non-compactness is
as good as compactness for our purposes.
Let us now make this precise. Suppose we have a sequence of τn → τ ≥ 0
together with curves ˜vn : DR → R × M satisfying the boundary value problem
(5.2). We need to establish this sequence must have uniformly bounded gradient
and hence a subsequence converges in C ∞ loc
on the interior of the disk. We also
need to establish a bound on ||fn||L∞ to show that the limiting disk satisfies the
boundary condition.
A priori bounds on |f|.
We will first establish an a priori bound on the amount by which we flow the
boundary in (5.2). We will establish this by establishing that the f we obtain is
of bounded variation. This, combined with the constraint that f(R) = 0 gives
us an L ∞ bound on f. We note that this result relies on the fact that we may
estimate the Maslov index of the boundary condition in terms of the orders of
the zeros of πλ T u on DR.
77

Lemma 5.3.1. Let (˜v, τ) be a solution to (5.2) for some value of 0 < τ ≤ 1, with
˜v = (b, v). We lift β : ∂DR → ∂DR to a map β : R → R with β(t+1) = β(t)+1.
Then,
• The function β satisfies ˙ β > 0,
• There is an a priori bound on the L ∞ norm of f, independent of τ, b and
v. This means that any solution of (5.2) has its boundary lying a finite
part {φT (u(z)) | z ∈ ∂DR , |T | ≤ C} ⊂ L
Proof. The first item follows immediately since the Maslov index of L is given
by twice the sum of the order of interior zeros of πλ T v plus the sum of the orders
of the boundary zeros. Now, πλ T v satisfies a first order Cauchy-Riemann type
PDE [23]. Thus, its zeros occur with positive multiplicity. For our original
curve, πλ T u is non-vanishing, so it follows that πλ T v must be non-vanishing.
In particular, πλ T v cannot have a zero at the boundary. Hence, ˙ β > 0.
The second result follows since b(z) is subharmonic. Indeed, we have that if
f : Σ → R is a smooth function and j a complex structure on Σ inducing the
orientation ds ∧ dt on Σ, then we define ∆jf by
−d(df ◦ j) = ∆jf ds ∧ dt.
The operator ∆j is a uniformly elliptic operator without any zero order term.
In particular, the weak and strong maximum principles apply. Our case is
particularly simple since the complex structure j = i on the disk. Then, ∆i =
∆ = ∂2
∂s2 + ∂2
∂t2 , the standard Laplacian. We note that b is subharmonic since
−d(db ◦ jρ) = v∗dλ ≥ 0.
We will show an a priori bound on the normal derivative of b at the boundary.
In order to do this, we need to have a zero boundary condition, which b
does not satisfy. To get around this, we will use the following well–known PDE
trick (pointed out to me by Casim Abbas).
We solve the following boundary value problem
◦
∆δ = 0 on DR
δ(z) = a(β(z)) on ∂DR
Consider now the function b − τ δ which satisfies
◦
DR
∆(b − τ δ) ≥ 0 on
b − τ δ = 0 on ∂DR
78

With ν denoting the outer normal vector along ∂DR we obtain
∂
(b − τ δ) > 0.
∂ν
Hence, for any point z ∈ ∂DR
∂DR
∂b ∂
(z) ≥ (τδ)(z) ≥ κ
∂ν ∂ν
for a suitable constant κ (possibly negative). By virtue of Poisson’s formula, we
see that the normal derivative of δ may be bounded from below by a constant
(depending only on R) times ||a||L ∞ (∂DR).
If we write z = e s+it , we obtain that z ∈ ∂DR is of the form z = Re 2πit . In
this case, up to a constant factor, ∂
∂ν
From this, we obtain :
= ∂
∂s .
∂b
(z)
∂ν
=
∂
∂s b(z)
= λv(z)[ ∂
∂t v(z)]
= ft(z) + λu(β(z))[ut(β(z))]βt(z)
= ft(z) − as(β(z))βt(z)
ft(z) = ∂
b(z) + as(β(z))βt(z)
∂ν
≥ κ − βt(z)||a(z)||L∞ This uses the fact that βt > 0.
Let now
S± := {t ∈ S 1 | ± ∂tf ≥ 0}.
Then, 0 =
∂DR ∂tf dt =
S+ ∂tf +
S− ∂tf.
We obtain
0 ≥ ft(t)dt ≥ κ|S−| − ||a||L∞
βt(t) ≥ κ − ||a||L∞ S−
because β is monotone increasing. From this we conclude that f has bounded
variation. Since f(R) = 0, we obtain a C 0 bound.
This bound depends only on R and ||a||L ∞ (DR).
79

Interior gradient bounds
We have now established an a priori bound on the amount we flow the boundary.
Thus, a sequence of solutions to the boundary value problem (5.2) that converge
in C∞ loc on the interior of the disk will have a subsequence converging to a solution
of the boundary value problem (5.2).
We now suppose we have a sequence of solutions to (5.2), ˜vn, corresponding
to τn → τ. If we have gradient bounds, a subsequence converges to a solution
to (5.2) for τ. If we do not have gradient bounds, we can find a sequence of
points zn so that |∇˜vn(zn)| =: Rn → ∞.
We introduce a slight variant of the E energy introduced by Hofer [21] to
study pseudoholomorphic curves in symplectizations of contact manifolds. We
recall that Hofer’s energy is given by
E(ũ) = sup{ ũ ∗ d(φλ)|φ : R → [0, 1] with φ′ ≥ 0}.
In the following analysis, we will use a variant of this energy in which we restrict
the class of φ that we allow. We will take φ so that
φ ′ = 0 on ∪n∈Z [(4n − 1)||a||L ∞ (∂DR), (4n + 1)||a||L ∞ (∂DR)].
We observe that this energy is essentially the same energy as used by Hofer,
but instead of being invariant under the action of R translation, it is invariant
under a discrete subgroup of translations. Nevertheless, invariance under
this discrete subgroup is sufficient to establish all of the existing results about
finite energy curves. In particular, finiteness of this energy implies the same
compactness results as the finiteness of the standard energy. In the following,
we will denote this variant of the energy by ER, to emphasize the fact that the
definition of the energy depends on R, the radius of the disk we take.
For any curve that satisfies our boundary condition on the disk, we have
that ER(˜v) ≤
∂DR v∗dλ by Stokes theorem and the fact that the φ in the
definition of the energy vanish on τa(z)|∂DR for τ ∈ [0, 1]. Since this energy is
finite, any interior bubbling gives a finite energy plane, which is then asymptotic
to a periodic orbit of action no greater than
C u∗dλ, in contradiction to our
assumption.
Boundary bubbling
We are now left with the case of bubbling at the boundary. We will show that
this implies that we bubble off a solution to our boundary value problem (5.2)
with τ = 0. Since proving the existence of such a curve was the goal of our
80

construction, this “short-circuits” the homotopy. The proof of this result relies
on our a priori bound on how far we flow in the immersed Lagrangian L.
Suppose ˜vn : DR → R × M are a sequence of solutions to (5.2) so that
we have a sequence of points zn → ∂DR so that Rn := |∇˜vn(zn)| → ∞. By
considering a subsequence, we may assume that zn → z∞ ∈ ∂DR.
Then, for n large, |zn − z∞| < 1/4. Let us now introduce a biholomorphic
change of coordinates :
We observe that
1
z − z∞
˜wn := ˜vn(− ).
z + z∞
where z ′ n = −z∞ ·
2 |∇˜vn(zn)| ≤ |∇ ˜wn(z ′ n)| ≤ 2|∇˜vn(zn)|
zn−1
zn+1
. Thus we may assume that we are working with
pseudoholomorphic half-planes on H = {z ∈ C | ℜ(z) ≥ 0} and a sequence of
points zn → ∂H, with |zn| ≤ 1.
By Hofer’s Lemma, we have a sequence ɛn > 0, with ɛn → 0 so that ɛnRn →
∞ and so that for |z − zn| < ɛn, we have |∇ũn(zn)| ≤ 2Rn. Let δn = d(zn, ∂H).
Then, we must have δnRn → c < ∞. Indeed, if δnRn → ∞, then we could
rescale along the disks D(zn, δn/2) and obtain interior bubbling.
Let ζn ∈ D(zn, ɛn) ∩ iR. Let ρn = ɛn − δn. Then, D(ζn, ρn) ⊂ D(zn, ɛn). We
may take a subsequence along which ɛn > 2δn. Then, zn ∈ D(ζn, ρn).
We consider now the following rescaling :
˜Wn := ˜wn( z
Rn
+ ζn) for |z| ≤ ρnRn and ℜ(z) ≥ 0
then we obtain that |∇ ˜ Wn(z)| ≤ 2 for |z| ≤ ρnRn. Furthermore,
|∇ ˜ Wn(Rn(zn − ζn))| = 1.
Since |Rn(zn − ζn)| ≤ Rnδn → c < ∞, we may take a subsequence along
which Rn(zn − ζn) converges. Let us denote the limit by ζ∞. We note that
|ζ∞| ≤ c.
By construction, the sequence ˜ Wn has a uniform gradient bound. Thus, a
subsequence converges to a pseudoholomorphic half-plane ˜ W = (C, W ) : H →
R × M.
Since the sequence ˜ Wn has a uniform gradient bound, for each k ≥ 1 and
for each compact K ⊂ H, we have uniform C k bounds. Let b be the bound for
C 2 (D(ζ∞, 3c)). We then have
|∇ ˜ Wn(ζ∞)| ≥ |∇ ˜ Wn(Rn(zn − ζn))| − sup(D 2 ˜ W )|ζ∞ − Rn(zn − ζn)|
≥ 1 − b|ζ∞ − Rn(zn − ζn)|,
81

Thus, |∇ ˜ W (ζ∞)| ≥ 1, so the curve is non-constant.
Let us write ˜vn = (bn, vn), ˜wn = (cn, wn), ˜ Wn = (C, W ) and ˜ W = (C, W ).
Then, we verify the boundary condition satisfied by ˜ W . For any point it ∈ ∂H,
we have
C(it) = lim Cn(it) = lim cn( it
Rn
+ ζn)
1 − it/Rn − ζn
= lim bn(z∞
)
1 + it/Rn + ζn
1 − it/Rn − ζn
= lim τna(z∞
)
1 + it/Rn + ζn
1 − ζ∞
= τ∞a(z∞ )
1 + ζ∞
= constant.
By shifting our curve by means of the R action, we may take the constant to
be zero.
We will now show that ˜ W , defined on a half-plane, has a removable singularity
at infinity. This will then show that ˜ W may be extended to a smooth
map on the disk, with boundary in L. This resulting disk will be non-constant
and thus will be the renormalized disk we are seeking.
Let us consider ˜ W with the half-plane re-parametrized as a strip. Thus, we
let ˜ W ′ (z) = ˜ W (ie 2π(s+it) ). Even though we had a bound on the gradient of ˜ W ,
we no longer have a bound on the gradient of ˜ W ′ . Let us suppose that the
gradient of ˜ W ′ is unbounded. Then, there is a sequence of points (sk, tk) in the
strip, along which |∇ ˜ W ′ | is unbounded. We have that sk → ∞ and may take a
subsequence along which tk → t∞. As before, we have no interior bubbling since
the resulting pseudoholomorphic plane would be asymptotic to a periodic orbit
we have assumed does not exist. Thus, we may only have boundary bubbling.
We consider now the case of boundary bubbling. After re-scaling, we obtain
a non-constant pseudoholomorphic half-plane, ˜ W ′′ = (B ′′ , W ′′ ). This half-plane
maps the boundary ∂H to {0}×L and has bounded gradient. Furthermore, the
half-plane satisfies
H W ′′∗ dλ = 0. Thus, B ′′ is harmonic on the half-plane with
bounded gradient. We obtain, therefore, that B ′′ (s, t) = cs for some c. Hence,
W ′′ (s, t) = x(ct), where x(t) is a trajectory of the Reeb vector field. We know,
however, that W ′′ has finite energy (in our modified sense) and thus c = 0 and
W ′′ is constant. This contradiction establishes that our original strip W ′ has
bounded gradient.
We will now show that ˜ W ′ has a removable singularity at +∞. Let us
introduce ˜ W ′ k (z) := ˜ W ′ (z − sk) for some sequence sk → ∞. This sequence
has uniform gradient bounds since W ′ does. Hence, ˜ W ′ k converge in C∞ loc to
82

a pseudoholomorphic strip, ˜ W ′′′ . Observe that ˜ W ′′′ = (B ′′′ , W ′′′ ) maps the
boundary of the strip to {0} × L and has 0 dλ area. Hence, B ′′′ is harmonic.
Furthermore, B ′′′ has bounded gradient. Hence, B ′′′ (s, t) ≡ 0. Thus ˜ W ′′′ is
constant.
We let
˜V (z) = (B(z), V (z)) : {z ∈ C | |z| ≤ 1 and z = 1} −→ R × M
z ↦→ ˜
1 + z
W .
1 − z
By our previous work, we have that B(z) is bounded along the boundary, with
boundary in a Lagrangian submanifold, {0} × L. Thus, by Gromov’s removable
singularity theorem, it follows that ˜ V may be extended to a smooth pseudoholomorphic
map with boundary in {0} × L.
We now observe that
V ∗
dλ ≤ u ∗ dλ.
D
DR
By Stokes theorem, since V (∂D) ∈ L, we have
V ∗
dλ = deg(V |∂D) ·
DR
u ∗ dλ.
Finally, since B = 0 on ∂D, E( ˜ V ) =
D V ∗ dλ (this energy is the standard,
Hofer energy). The map ˜ V is non-constant, so this energy must be positive.
Thus we must have deg(V |∂D) = 1. This shows that ˜ V is a solution to the
boundary value problem (5.2), for τ = 0. Thus, if we have boundary bubbling
along our sequence, we immediately obtain a renormalized solution.
Conclusion of the proof
Let us assume for now that we have proved the local existence result, that if we
have a solution of (5.2) for τ = τ0, then there exist nearby solutions for τ close
to τ0. (We prove this in the following section, 5.3.3).
We then have that for any disk DR, we may find a renormalized solution :
⎧
⎪⎨
⎪⎩
˜vR = (bR, vR) : DR → R × M
πλ T v ◦ i = J(v) ◦ πλ T v
v ∗ λ ◦ i = db
b(z) = 0 for z ∈ ∂DR
v(z) = φf(z)(u(β(z))) for z ∈ ∂DR
where f : ∂DR → R
and β : ∂DR → ∂DR of degree 1
with the constraint, f(R) = 0
83
(5.4)

We observe now that the standard Hofer energy, E(˜vR) =
BR u∗d λ, by
Stokes Theorem.
We take a sequence Rk → ∞. Let ˜vk = ˜vRk . For each k, E(˜vk) ≤ C. We
have one of two possibilities, after taking subsequences as necessary :
(a) For each n, there exists a bound Mn > 0 so that for all k and for all
|z| ≤ min{Rn, Rk}, we have |∇vk(z)| ≤ Mn. In this case, after a diagonal
subsequence argument, we obtain that, after translation in the symplecti-
zation direction, ˜vk converge in C ∞ loc
to a finite energy pseudoholomorphic
plane. This plane must then be asymptotic to a periodic Reeb orbit of
period no greater than C. This contradicts our assumption.
(b) There exists a sequence of points zk so that |∇˜vk(zk)| → ∞ as k → ∞.
We may now apply the usual bubbling off argument to obtain a finite
energy plane, ˜w, with E( ˜w) ≤ C. This is again a contradiction to our
assumption.
Theorem 5.1.1 follows from the contradictions in (a) and (b).
5.3.3 Fredholm theory
In this section, we will establish the following result :
Proposition 5.3.2. Suppose the boundary value problem (5.2) has a solution
for some τ0 > 0. Then there exist solutions for τ sufficiently close to τ0.
We first make the observation that it suffices to consider τ0 = 1. Indeed, if
we have a solution ˜vτ0, for τ0, then we may “re-start” our homotopy by taking
ũ0 = ˜vτ0.
Thus, in this section we will introduce a suitable functional analytic set-up,
and prove that our problem is Fredholm and surjective at the initial solution
for τ = 1, ũ. We will also show that the kernel of of the linearized operator
includes an element which projects non-trivially on ∂ . Thus, we will conclude
∂τ
that the solutions exist for nearby values of τ.
This work is very similar to the case in a closed symplectic manifold (as in
[33]). We wish to consider all nearby curves that satisfy the boundary condition
as elements of a Banach manifold. Such curves may be represented as graphs
over ũ0 by using the flow of ∂t (the generator of the symplectization direction)
and the flow of X (the Reeb vector field), together with a re-parametrization
of the domain. This allows us to identify a sufficiently small neighbourhood of
our original curve with a small neighbourhood of the zero section in the bundle
E := C ⊕ T D → D. We may now pull the almost complex structure ˜ J back to
84

a complex structure on E, ¯ J. We define the Sobolev space W 1,p (E) of sections
of this bundle. We now define a Banach manifold
B := {u ∈ W 1,p (E) | u satisfies the boundary conditions}.
Over this Banach manifold, we put a Banach space bundle E, whose fibre over
a section v is given by
Ev = Hom 0,1 ( T D, v ∗ E)
where Hom 0,1 refers to the bundle of i– ¯ J antiholomorphic real linear homomorphisms.
The (non-linear) Cauchy–Riemann operator,
v ↦→ dv − ¯ J ◦ dv ◦ i
is a smooth section of this Banach space bundle. Its linearization at the zero
section will turn out to be a linear Cauchy–Riemann operator. This is then
Fredholm, with the index determined by the boundary data (as given by the
Riemann-Roch theorem).
Finding holomorphic sections of a vector bundle
Our goal is to show that our problem may be identified with a non-linear problem
whose linearization at our initial curve is of linear Cauchy-Riemann type. We
will thus construct a Banach manifold consisting of maps from the disk that
satisfy the boundary condition. On this, we will put the Banach space bundle
whose fibre over a maps v consists of antiholomorphic sections in v∗T W (where
W is an almost complex manifold we will specify shortly). The non-linear
operator will then be a smooth section of this bundle.
For clarity of notation, we will work with R = 1. This covers the general
case, since we may precompose ũ with the map z ↦→ z/R.
Our first step is to introduce convenient coordinates. To this end, we will
observe that since πλ T u = 0 on D, we have that ∂ and X span the normal
∂a
bundle of ũ in ũ∗ (R × M). Notice that we have that ũ is immersed. Thus, we
may write a C1 close curve as a graph in the normal bundle of ũ, with possibly
a different parametrization of the domain. This means we then have
Lemma 5.3.3. A solution ˜v to (5.2), sufficiently C 1 close to ũ, may be written
as v(z) = φf(z) ◦ u(β(z)), where f : D → R and β : D → D. Moreover, β
restricts to a degree 1 map on the boundary and β(z) = z for all z ∈ Γ.
This lemma allows us to introduce the coordinates we will use for our Fredholm
theory :
Ψ : R × R × D → R × M
(ρ, σ, ξ, η) ↦→ (ρ + a(ξ, η), φσ ◦ u(ξ, η))
85

We note that Ψ is an immersion.
Let us introduce ¯ J on R × R × D by T Ψ ◦ ¯ J = ˜ J ◦ T Ψ. We thus transformed
our problem (5.2) into a problem of finding pseudoholomorphic disks into the
almost complex manifold W = R × R × D with the almost complex structure
¯J : (in the following, we identify C with R 2 )
w = (f, h) : D → C × D,
T w ◦ i = ¯ J ◦ T w on D
h|∂D : ∂D → ∂D is a degree 1 map,
f1(z) = (τ − 1)a(z) for z ∈ ∂D
f2(1) = 0
(5.5)
We observe that for τ = 1, we have the solution (w, c) given by w : z ↦→
(0, 0, z).
We now put then a metric on D for which the boundary is totally geodesic.
To do this, we consider a metric that near the boundary is of the form ds 2 ⊕ dt 2
where z = e 2π(s+it) . Outside of a neighbourhood of the boundary, we take the
metric to be the standard metric on the disk.
We take now the exponential map from this metric. Any map h : D → D,
mapping boundary to boundary, sufficiently C 0 close to the identity map may
be represented as h(z) = exp z(H(z)), where H : D → T D is a vector field on
D, with H(z) ∈ izR for z ∈ ∂D.
Let us now define the bundle
E = C ⊕ T D → D.
We fix a one parameter family of totally real subbundles at the boundary given
by
ℓτ|z = ((1 − τ)a(z) + iR, izR) for |z| = 1.
To a section of this bundle, we may define a Sobolev norm. Indeed, we
may identify a section of E with a map (s1, s2) : D → C ⊕ C. On this, we
put the standard Sobolev norms W 1,p . We denote the space of these sections
by W 1,p (E). We observe that any two choices of trivialization of E induce
equivalent norms.
We are primarily interested in certain special subspaces of W 1,p , representing
variations of curves that satisfy our boundary conditions. We note that we
have two different types of boundary condition : we have a one parameter
family of totally real boundary conditions, and we have a pointwise constraint.
We will thus find it useful to introduce four spaces : we will consider the one
parameter family of totally real boundary conditions, with and without the
pointwise constraint; we will also consider a boundary condition in a fixed totally
86

eal boundary condition (at τ = 1) with and without the pointwise constraint.
The first such space we denote by W 1,p
ℓτ . This is the space of W 1,p sections s such
that there exists a τ = τ(s) so that s|∂D ∈ ℓτ. The next such space we denote by
W 1,p
1,p
ℓτ ,pt . This is the co-dimension one subspace of Wℓτ such that a section (s1, s2)
has Im(s1)(1) = 0. The third space we introduce is another codimension one
subspace of W 1,p
, corresponding to the sections for which τ = 1. We denote this
ℓτ
by W 1,p
ℓ , and is the space of sections s with s|∂D ∈ iR⊕izR. Finally, we consider
the space W 1,p
ℓ,pt
, consisting of all sections in W 1,p
ℓτ ,pt such that s(z) ∈ ℓ1 = iR⊕izR
for z ∈ ∂D. This corresponds to the space of sections that satisfy our pointwise
constraint.
We have that for ||s|| W 1,p < ɛ0 sufficiently small, we have that ||s||L∞ is
small, thus the map (s1, s2) ↦→ (s1, expz s2) is injective.
We define now the following Banach manifold :
Bℓτ = {z ↦→ (s1(z), expz s2(z)) | s ∈ W 1,p
(E), with ||s|| < ɛ0}.
ℓτ
and similarly Bℓ, Bℓτ ,pt and Bℓ,pt. We note that each of these is a submanifold
of Bℓτ .
Over this large Banach manifold, we now put the Banach space bundle
F → Bℓτ so that the fibre over any u ∈ Bℓτ is given by
Fu = Hom 0,1 ( T D, u ∗ T W ).
On this bundle, we may put the L p norm, similarly to our previous construction.
We define now the non-linear operator, ¯ ∂ ¯ J, whose zeros are pseudoholomorphic
curves.
¯∂ ¯ J : Bℓτ −→ E
u ↦→ du − ¯ J ◦ du ◦ i.
(5.6)
We recall that ¯ ∂ ¯ Ju0 = 0, where u0(z) = (0, 0, z). We note that the zeros of
¯∂ ¯ J restricted to the space Bℓτ ,pt are precisely the solutions of (5.2) near to our
original curve.
The fact that the various B are Banach manifolds and that F is a Banach
space bundle over them follows from the work of Eliasson [14]. We note that ¯ J
is smooth, and thus it follows (again by [14]) that ¯ ∂ ¯ J is a smooth section of F.
In particular, then, it is C 1 .
We consider now the problem of finding zeros of the operator restricted to
the space Bℓ. In other words, we are working over the space of maps that
satisfy the boundary condition for a fixed totally real boundary condition. By
[33, Prop. 3.1.1.], the linearization of ¯ ∂ ¯ J at a map v, which we denote Lv, is a
Cauchy-Riemann type operator.
Lv : W 1,p
ℓ (v ∗ E) −→ L p (Hom 0,1 ( T ˙ Σ, v ∗ E)).
87

In particular, when we linearize the operator at the zero section, we obtain
a Cauchy-Riemann type operator L0 := D( ¯ ∂ ¯ J)(0, z)| W 1,p
ℓ (E)
L0 : W 1,p
ℓ (E) −→ L p (Hom 0,1 ( T D, E)).
By the Riemann-Roch theorem [33, Theorem C.1.10], we have then that this
operator is Fredholm, and its index is given by :
Ind(L0) = 2 + µ(ℓ) = 2 + 2 = 4.
We have that L0 is the restriction of D( ¯ ∂J)(0, ¯ z) to the space W 1,p
ℓ from the
space W 1,p
1,p 1,p
. We have that W ℓτ ℓ ⊂ W has co-dimension one. Thus, the full
ℓτ
linearization of ¯ ∂ ¯ J at the zero section,
D( ¯ ∂ ¯ J)(0, z) : W 1,p
ℓτ (E) −→ Lp (Hom 0,1 ( T D, E))
is Fredholm, and its index is Ind(L0) + 1 = 5.
Finally, we wish to consider D( ¯ ∂J)(0, ¯ z), restricted to the subspace corresponding
to subspace enforcing the pointwise constraint, W 1,p
ℓτ ,pt . Let us call this
restricted operator L0. This is then L0 : W 1,p
ℓτ ,pt (E) −→ Lp (E). By virtue of the
fact that W 1,p
ℓτ ,pt
and its index is given by
(E) is codimension one in W 1,p
ℓτ (E), we have that L0 is Fredholm,
Ind(L0) = Ind(D( ¯ ∂ ¯ J)(0, z)) − 1 = 4.
The same argument applies to the restriction of D( ¯ ∂J)(0, ¯ z) to W 1,p
ℓ,pt , where
the index is then 3.
Transversality
We now must prove transversality for L0. In other words, we must show that
the kernel of L0 is four dimensional. Furthermore, to show that we have a family
that depends on the parameter τ, we must show that the kernel of L0 projects
non-trivially on the τ direction.
Once this is established, by the implicit function theorem, we will have a
four parameter family of solutions to the problem (5.5) near our original curve.
Furthermore, we will have that one of these parameters may be taken to be τ.
This will establish then Lemma 5.3.3.
In order to establish this result, we will show that
L0 := D( ¯ ∂ ¯ J)(0, z)| W 1,p
ℓ,pt (E)
88

is surjective. The result will then follow, since L must then be surjective, and
thus its kernel is four dimensional. This then must project non-trivially on the
τ direction.
In order to show that we have transversality for L0, we must pay more
attention to the form of the linearization of the operator. To this end, we will
use the fact that we have global holomorphic coordinates on D ⊂ C, given by
z = x + iy, and we also have an isomorphism from Hom 0,1 ( T D, E) to E by the
map φ ↦→ φ ∂
∂x .
We also must study the behaviour of our almost complex structure ¯ J. Let
us put coordinates on R × R × D by (ρ, σ, x, y). (We recall that we let a be the
coordinate on the R factor in R × M. We also have ũ = (a, u) is our original
curve, and J is the complex structure on ker λ ⊂ T M.) We obtain :
where
Let
Ψ∗
Ψ∗
Ψ∗
∂
∂ρ
∂
∂x
∂
∂y
=
∂
∂a
∂
Ψ∗
∂σ = X(φσ ◦ u(z))
=
∂
ax
∂a + T φσ ◦ ux
=
∂
ax
∂a + λ(ux)X + ν1
=
∂
ay
∂a + λ(uy)X + ν2.
ν1(σ, z) := T φσ ◦ πλ ux, ν2(σ, z) := T φσ ◦ πλ uy.
B =
ax
ay
.
λ(ux) λ(uy)
By the fact that ũ is pseudoholomorphic with respect to ˜ J, we obtain that B
commutes with J0, the standard complex structure on R 2 identified with C.
Let ˆ J(σ, z) be the matrix representing J(z) with respect to the basis ν1(σ, z)
and ν2(σ, z). We note that ˆ J is an almost complex structure on C. We have
that ˆ J(0, z) = J0, the standard complex structure.
Then, we have, with respect to the basis ∂
∂ρ
by the matrix ¯ M,
Let N(σ, z) = ˆ J(σ, z) − J0.
, ∂
∂σ
, ∂
∂x
J0 ¯M(σ,
B J0 −
x, y) =
ˆ
J(σ, z)
.
0 J(σ, ˆ z)
89
, ∂
∂y , that ¯ J is represented

It follows that the non-linear operator, acting on the section (f, h) =
(f1, f2, h1, h2) : D → C ⊕ T D, may be expressed as :
∂xf + i∂yf − B(expz h(z))N(f2, expz h(z))∂y(exp
(f, g) ↦→
z(h(z)))
∂xh + i∂yh + N(σ, expz(h(z)))∂y(expz(h(z))) We now linearize this operator at the zero section. This gives us a linearized
operator of the form
¯∂J0F + P (z)F
(F, G) ↦→ ¯∂J0H + Q(z)F
where P and Q are matrix valued functions.
We first consider all pairs (0, H) in the kernel of this operator. Hence, we
have that H : D → C is a holomorphic function (classically), with H(z) ∈ izR
for |z| = 1. We then have a three (real) parameter family of such H.
We now consider a pair (F, H) in the kernel of the operator. It follows then
that ¯ ∂J0F + P (z)F = 0. Furthermore, F (z) ∈ iR for |z| = 1. This is a totally
real boundary condition of Maslov index 0. Thus, if F is not identically 0, F
cannot vanish anywhere. However, we have that F (1) = 0. Thus, F ≡ 0.
We then have that the kernel of L is three dimensional. The Fredholm index
of L is also three, so it follows that L is surjective. The result now follows.
5.4 Future work
The present work on the behaviour of infinite E energy, but finite contact area,
pseudoholomorphic planes is a beginning of further work with the compactification
of the space of solutions to the generalized pseudoholomorphic curve
equations with harmonic form as introduced by Hofer in [3].
The technical constraint that πλ T u never vanish was so that ∂
∂a
and X
spanned the normal bundle to the image of ũ. In order to deal with the zeros
of πλ T u, we will study a problem on ˙ D = D \ Γ. We will see this as a Riemann
surface with punctures. The problem will then become more difficult,
analytically speaking.
The current work, along with some examples, indicates that these infinite
energy, finite contact area planes are asymptotic to periodic orbits in a suitable
(weak) sense. To study this, we will consider the renormalization of long annuli
with contact area below the energy threshold from Theorem 5.1.1. The goal
is to develop a theory analogous to the work in [26], where a similar theory
is developed for finite energy curves. In the case of annuli, the compactness
problem becomes more difficult. This is ongoing project with Casim Abbas
and Helmut Hofer. The ultimate goal of this project is to develop a good
compactification for the space of Hofer’s generalized pseudoholomorphic curves.
90

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