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The corresponding statement for jumping conormal boundary conditions is the following:
6.5. Proposition. Let X :�1 ∩À×T ∗ Q → TT ∗ Q be a smooth vector field such that X(z, q, p)
grows at most polynomially in p, uniformly in (z, q). Let u : Cl(�1 ∩À) \ {0} → T ∗ Q be a smooth
solution of the equation
such that
If u has finite energy,
∂Ju(z) = X(z, u(z)) ∀z ∈ Cl(�1 ∩À) \ {0},
u(s) ∈ N ∗ Q0 ∀ s > 0, u(s) ∈ N ∗ Q1 ∀ s < 0.
��1∩À|∇u| 2 dsdt < +∞,
then u extends to a continuous map on the closed half-disk Cl(�1 ∩À).
Proof. The energy is invariant with respect to conformal changes of variable. Therefore, it is
enough to apply Proposition 6.4 to the map v(z) = u(z 2 ), with z ∈�1 ∩À+ .
6.3 Proof of Proposition 3.13
Let x1 ∈ P(H1), x2 ∈ P(H2), and z ∈ P(H1#H2) be such that
µ Λ (x1) + µ Λ (x2) − µ Λ (z) = n, (162)
so that the manifold M Υ GE (x1, x2; z) is one-dimensional. By standard arguments, Proposition 3.13
is implied by the following two statements:
(i) for every y ∈ P Θ (H1 ⊕ H2) such that
µ Θ (y) = µ Λ (z) = µ Λ (x1) + µ Λ (x2) − n,
and every pair (u1, u2) with u1 ∈ ME(x1, x2; y) and u2 ∈ MG(y, z), there is a unique
connected component of M Υ GE (x1, x2; z) containing a curve α ↦→ (α, uα) which - modulo
translations in the s variable - converges to (+∞, u1) and to (+∞, u2);
(ii) for every u ∈ M Λ Υ (x1, x2; z), there is a unique connected component of M Υ GE (x1, x2; z)
containing a curve α ↦→ (α, uα) which converges to (0, u).
The first statement follows from standard gluing arguments. Here we prove the second statement,
by reducing it to an implicit function type argument. At first the difficulty consists in a parameter
dependence of the underlying domain for the elliptic PDE. Using the special form of the occuring
conormal type boundary conditions and a suitable localization argument we equivalently translate
this parameter dependence into a continuous family of elliptic operators with fixed boundary
conditions.
If (α, v) ∈ M Υ GE (x1, x2; z), we define u : Σ =Ê×[0, 1] → T ∗ M 2 as
and the Hamiltonian K onÌ×T ∗ M 2 by
u(s, t) := (−v(s, −t), v(s, t)),
K(t, x1, x2) := H1(−t, −x1) + H2(t, x2).
By this identification, we can view the space M Υ GE (x1, x2; z) as the space of pairs (α, u), where
α > 0 and u : Σ → T ∗ M 2 solves the Floer equation
∂J,K(u) = 0, (163)
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