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Since we have applied a conformal rescaling, the energy of vh is uniformly bounded, so (vh) converges
up to a subsequence to some J-holomorphic map v in the C0 loc ([0, +∞[×[0, +∞[, T ∗M4 )
topology (more precisely, we have C∞ loc convergence once the domain [0, +∞[×[0, +∞[ is transformed
by a conformal mapping turning the portion near the boundary point (1, 0) into a neighborhood
of (0, 0) in the upper-right quarterÀ+ =]0, +∞[×]0, +∞[). The J-holomorphic map has
finite energy, so by removal singularities it has a continuous extension at ∞ (again, by Proposition
6.4 together with a suitable conformal change of variables). By (203), (204), and (205) it satisfies
the boundary conditions
π ◦ v(0, t) = (c(0), c(0)) for t ≥ 0, (208)
v(s, 0) ∈ N ∗ ∆M×M for 0 ≤ s ≤ 1, (209)
v(s, 0) ∈ N ∗ ∆ Θ M
for s ≥ 1. (210)
Since the boundary conditions are of conormal type and the Liouville one-form η vanishes on
conormals, we have
|∇v| 2 ds dt = v ∗ (ω) = v ∗ (dη) = dv ∗ �
(η) = v ∗ (η) = 0,
�À+
�À+
�À+
�À+
so v is constant. By (208), π ◦ v = (c(0), c(0)). In particular,
Since
the thesis follows.
∂À+
lim
h→+∞ π ◦ vh(s, 0) = (c(0), c(0)) uniformly in s ∈ [0, 1].
π ◦ vh(s, 0) = π ◦ ũh(αhs, 0) = (π ◦ uh(αhs, 0), π ◦ uh(s, 1)) = (dh(s), dh(s)),
Localization. We fix a positive number A, playing the role of the upper bound for the action.
Then the union of all spaces of solution M K 0 (γ1, γ2; x), where γ1 ∈ P Λ (L1), γ2 ∈ P Λ (L2), and
x ∈ P Θ (H1 ⊕ H2) satisfy the index identity (193) and the action estimates
ËL1(γ1) +ËL2(γ2) ≤ A,�H1⊕H2(x) ≤ A, (211)
is a finite set. Let us denote by qi, i ∈ {1, . . ., m}, the points in M such that π ◦ ui(0, 0) =
π ◦ ui(0, 1) = (qi, qi) for some u in the above finite set. We choose the indexing in such a way that
the points qi are pair-wise distinct, and we fix a positive number δ such that
Bδ(qi) ∩ Bδ(qj) = ∅, ∀ i �= j.
We may assume that the positive constant δ chosen above is so small that Bδ(qi) ⊂ M is diffeomorphic
toÊn . Lemma 6.7 implies the following localization result:
6.8. Lemma. There exists a positive number α(A) such that for every α ∈]0, α(A)], every γ1 ∈
P Λ (L1), γ2 ∈ P Λ (L2), x ∈ P Θ (H1 ⊕ H2) satisfying the index identity (193) and the action
bounds (195), each solution u ∈ M K α (γ1, γ2; x) satisfies
for some i ∈ {1, . . ., m}.
π ◦ u([0, α] × {0}) = π ◦ u([0, α] × {1}) ⊂ B δ/2(qi) × B δ/2(qi),
The chain homotopy. Since the Grassmannian of subspaces of some given dimension inÊn is
connected and since the δ-ball around each qi is diffeomorphic toÊn , there exist smooth isotopies
ϕij : [0, 1] ×Ê3n → Bδ(qi) 4 × Bδ(qj) 4 ⊂ M 8 , ∀i, j ∈ {1, . . .,m},
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