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For the remaining part of the argument leading to C∞ loc compactness of M Λ Υ (x1, x2; y) it is more
convenient to use the original definition of this solutions space and the smooth structure of ΣΛ Υ .
Then the argument is absolutely standard: If by contradiction there is no uniform C1 bound, a
concentration argument (see e.g. [HZ94, Theorem 6.8]) produces a non-constant J-holomorphic
sphere. However, there are no non-constant J-holomorphic spheres on cotangent bundles, because
the symplectic form ω is exact. This contradiction proves the C1 bound. Then the Ck bounds for
arbitrary k follow from elliptic bootstrap, as in [HZ94, section 6.4].
Other solutions spaces, such as the space M Ω Υ
for the traingle products, involve Riemann
surfaces with boundary, and the solutions take value on some conormal subbundle of T ∗ M. In
this case the concentration argument for proving the C 1 bound could produce a non-constant
J-holomorphic disk with boundary on the given conormal subbundle. However, the Liouville oneform
vanishes on conormal subbundles, so such J-holomorphic disks do not exist. Again we find
a contradition, leading to C 1 bounds and - by elliptic bootstrap - to C k bounds for every k.
6.2 Removal of singularities
Removal of singularities results state that isolated singularities of a J-holomorphic map with
bounded energy can be removed (see for instance [MS04, section 4.5]). In Proposition 6.4 below,
we prove a result of this sort for corner singularities. The fact that we are dealing with cotangent
bundles, which can be isometrically embedded into�N , allows to reduce such a statement to the
following easy linear result, where�r is the open disk of radius r in�, andÀ+ is the quarter plane
{Rez > 0, Im z > 0}.
6.3. Lemma. Let V0 and V1 be partially orthogonal linear subspaces ofÊn . Let u : Cl(�1 ∩À+ ) \
{0} →�n be a smooth map such that
for some p > 2, and
u ∈ L p (�1 ∩À+ ,�n ), ∂u ∈ L p (�1 ∩À+ ,�n ),
u(s) ∈ N ∗ V0 ∀ s > 0, u(it) ∈ N ∗ V1 ∀ t > 0.
Then u extends to a continuous map on Cl(�1 ∩À+ ).
Proof. Since V0 and V1 are partially orthogonal, by applying twice the Schwarz reflection argument
of the proof of Lemma 5.6 we can extend u to a continuous map
u :�1 \ {0} →�n ,
which is smooth on�1 \ (Ê∪iÊ), has finite L p norm on�1, and satisfies
∂u ∈ L p (�1).
Since p > 2, the L 2 norm of u on�1 is also finite, and by the conformal change of variables
z = s + it = eζ = eρ+iθ , this norm can be written as
� 0 � 2π
��1
|u(z)| 2 dsdt =
−∞
0
|u(e ρ+iθ )| 2 e 2ρ dθdρ.
The fact that this quantity is finite implies that there is a sequence ρh → −∞ such that, setting
ǫh := e ρh , we have
lim
h→∞ ǫ2 h
� 2π
0
|u(ǫhe iθ )| 2 dθ = lim
h→∞ e2ρh
� 2π
If ϕ ∈ C ∞ c (�1,�N ), an integration by parts using the Gauss formula leads to
=
��1
��ǫ
��1\�ǫ
〈u, ∂ϕ〉dsdt = 〈u, ∂ϕ〉dsdt + 〈u, ∂ϕ〉dsdt
h
h
��ǫ h
〈u, ∂ϕ〉dsdt −
��1\�ǫ h
0
�
〈∂u, ϕ〉dsdt + i
91
|u(e ρh+iθ )| 2 dθ = 0. (158)
∂�ǫ h
〈u, ϕ〉dz.