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to ũ0 uniformly on Σ, we may localize the problem and assume that M =Ên . More precisely, if
the projection of ũ0(z) onto M 4 is (q1, q2, q3, q4)(z), we construct open embeddings
Σ ×Ên → Σ × M, (z, q) ↦→ (z, ϕj(z, q)), j = 1, . . . , 4,
such that ϕj(z, 0) = qj(z) and D2ϕj(z, 0) is an isometry, for every z ∈ Σ (for instance, by
composing an isometric trivialization of q∗ j (TM) by the exponential mapping). The induced open
embeddings
Σ × T ∗Ên → Σ × T ∗ M, (z, q, p) ↦→ (z, ψj(z, q, p)) := � z, ϕj(z, q), (D2ϕj(z, q) ∗ ) −1 p � , j = 1, . . .,4,
are the components of the open embedding
Σ × T ∗Ê4n → Σ × T ∗ M 4 , (z, ξ) ↦→ (z, ψ(z, ξ)) := � z, ψ1(z, ξ1), . . . , ψ4(z, ξ4) � .
Such an embedding allow us to associate to any ũ : Σ → T ∗ M 4 which is C 0 -close to ũ0 a map
w : Σ → T ∗Ê4n =�4n , by setting
ũ(z) = ψ(z, w(z)).
Then ũ solves (167) if and only if w solves an equation of the form
D(w) := ∂sw(z) + J(z, w(z))∂tw(z) + G(z, w(z)) = 0, (172)
where J is an almost complex structure on�4n parametrized on Σ and such that J(z, 0) = J0
for any z ∈ Σ, whereas G : Σ ×�4n →�4n is such that G(z, 0) = 0 for any z ∈ Σ. Moreover, ũ
solves the asymptotic conditions (171) if and only if w(s, t) tends to 0 for s → ±∞. The maps
ψj(z, ·) preserve the Liouville form, so they map conormals into conormals. It easily follows that
the boundary condition (168) on ũ is translated into
w(s, 1) ∈ N ∗ ∆Ê2n ∀s ∈Ê. (173)
Moreover, ũ satisfies the boundary conditon (169) if and only if w satisfies
⎧
⎨ N
w(s, 0) ∈
⎩
∗∆Ê2n if s ≤ 0,
N ∗∆ΘÊn if 0 ≤ s ≤ 2α,
N ∗ (∆Ên × ∆Ên) if s ≥ 2α.
Similarly, ũ satisfies the boundary condition (170) if and only if w satisfies
�
∗ N ∆Ê2n if s ≤ 0,
w(s, 0) ∈
N ∗ (∆Ên × ∆Ên) if s ≥ 0.
(174)
(175)
The element u0 ∈ M Λ Υ (x1, x2; z) corresponds to the solution w0 = 0 of (172)-(175). By using the
functional setting introduced in section 5, we can view the nonlinear operator D defined in (172)
as a continuously differentiable operator
D : X 1,p
S ,V ,V ′(Σ,�4n ) → X p
S (Σ,�4n )
where S := {0}, V := (∆Ê2n, ∆Ên × ∆Ên), V ′ := (∆Ê2n), and p is some number larger than 2.
Since J(z, 0) = J0, the differential of D at w0 = 0 is a linear operator of the kind studied in section
5, and by the transversality assumption it is an isomorphism.
Consider the orthogonal decomposition
where
Ê4n = W1 ⊕ W2 ⊕ W3 ⊕ W4,
W1 := ∆ ΘÊn = ∆Ê2n ∩ (∆Ên × ∆Ên), ∆Ê2n = W1 ⊕ W2, ∆Ên × ∆Ên = W1 ⊕ W3,
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