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The Riemann surface ΣG is obtained from the disjoint union of two strips (Ê×[−1, 0]) ⊔ (Ê×
[0, 1]) by making the identifications:
�
− + (s, 0 ) ∼ (s, 0 )
for s ≥ 0.
(s, −1) ∼ (s, 1)
A holomorphic coordinate at (0, 0) is the one given by (36), and a holomorphic coordinate at
(0, −1) ∼ (0, 1) is:
�
2 ζ − i if Imζ ≥ 0,
{ζ ∈�|Reζ ≥ 0, |ζ| < 1} → ΣG, ζ ↦→
ζ2 (44)
+ i if Imζ ≤ 0,
We obtain a Riemann surface with two boundary lines and two strip-like ends (on the left-hand
side), and a cylindrical end (on the right-hand side). The global holomorphic coordinate z = s+it
has two singular points, at (0, 0), and at (0, −1) ∼ (0, 1).
Let H ∈ C ∞ (Ê/2�×T ∗ M) be defined by (40). Given x1 ∈ P Λ (H1), x2 ∈ P Λ (H2), y =
(y1, y2) ∈ P Θ (H1 ⊕ H2), and z ∈ P Λ (H1#H2), we consider the following spaces of maps. The
set ME(x1, x2; y) is the space of solutions u ∈ C ∞ (ΣE, T ∗ M) of the Floer equation
∂J,H(u) = 0,
satisfying the boundary conditions
� πu(s, −1) = πu(s, 0 − ) = πu(s, 0 + ) = πu(s, 1),
u(s, 0 − ) − u(s, −1) + u(s, 1) − u(s, 0 + ) = 0,
and the asymptotic conditions
∀s ≥ 0,
lim
s→−∞ u(s, t − 1) = x1(t), lim
s→−∞ u(s, t) = x2(t), lim
s→+∞ u(s, t − 1) = y1(t), lim u(s, t) = y2(t),
s→+∞
uniformly in t ∈ [0, 1]. The set MG(y, z) is the set of solutions u ∈ C ∞ (ΣG, T ∗ M) of the same
equation, the same boundary but for s ≤ 0, and the asymptotic conditions
lim
s→−∞ u(s, t − 1) = y1(t), lim
s→−∞ u(s, t) = y2(t), lim u(s, 2t − 1) = z(t),
s→+∞
uniformly in t ∈ [0, 1]. The following result is proved in section 5.10.
3.9. Proposition. For a generic choice of H1 and H2, the spaces ME(x1, x2; y) and MG(y, z)
- if non-empty - are manifolds of dimension
dimME(x1, x2; y) = µ Λ (x1) + µ Λ (x2) − µ Θ (y) − n, dimMG(y, z) = µ Θ (y) − µ Λ (z).
These manifolds carry coherent orientations.
The energy identities are now
� �Ê×]−1,1[
|∂su(s, t)| 2 ds dt =�H1(x1) +�H2(x2) −�H1⊕H2(y), (45)
for every u ∈ ME(x1, x2; y), and
� �Ê×]−1,1[
|∂su(s, t)| 2 ds dt =�H1⊕H2(y) −�H1#H2(z), (46)
for every u ∈ MG(y, z). As usual, they imply the following compactness result (proved in section
6.1).
3.10. Proposition. Assume that H1, H2 satisfy (H1), (H2). Then the spaces ME(x1, x2; y) and
MG(y, z) are pre-compact in C ∞ loc .
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