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y the argument of breaking Floer trajectories. Due to finite energy,
E(u) ≤ liminf
h→+∞ E(uh) ≤ËL1(γ1) +ËL2(γ2) −�H1⊕H2(x),
we find by removal singularities (Proposition 6.4) a continuous extension of u to the corner points
(0, 0) and (0, 1). By (195) and (196) we have
(u(0, 0), −u(0, 1)) ∈ N ∗ ∆ Θ M, π ◦ u(0, 0) = π ◦ u(0, 1) = c(0) = c(1).
It follows that the two components of the closed curve c coincide at the starting point, so they
describe a figure-eight loop, and u belongs to M K 0 (˜γ1, ˜γ2; ˜x). Since the latter space is empty
whenever
m Λ (˜γ1) + m Λ (˜γ2)) − µ Θ (˜x) < n,
the index assumption (193) together with (194) and (197) implies that ˜γ1 = γ1, ˜γ2 = γ2, and
˜x = x. We conclude that
u ∈ M K 0 (γ1, γ2; x).
We can also say a bit more about the convergence of (uh) towards u:
6.7. Lemma. Let dh : [0, 1] → M × M be the curve
dh(s) := π ◦ uh(αhs, 0) = π ◦ uh(αs, 1).
Then (dh) converges uniformly to the constant curve c(0) = c(1).
Proof. It is convenient to replace the non-local boundary conditions (189), (190) by local ones, by
setting
ũh : [0, +∞[×[0, 1/2] → T ∗ M 4 , uh(s, t) := (uh(s, t), −uh(s, 1 − t)).
Then ũh solves an equation of the form
∂ J, ˜ H (ũh) = 0,
for a suitable Hamiltonian ˜ H on [0, 1/2] × T ∗ M 4 , and boundary conditions
Then the rescaled map
solves the equation
with boundary conditions
π ◦ ũh(0, t) = (ch(t), ch(1 − t)) for 0 ≤ t ≤ 1/2, (198)
ũh(s, 0) ∈ N ∗ ∆M×M for 0 ≤ s ≤ αh, (199)
ũh(s, 0) ∈ N ∗ ∆ Θ M for s ≥ αh, (200)
ũh(s, 1/2) ∈ N ∗ ∆M×M for s ≥ 0, (201)
lim
s→+∞ ũh(s, t) = (x(t), −x(1 − t)). (202)
vh : [0, +∞[×[0, 1/(2αh)] → T ∗ M 4 , vh(s, t) = ũh(αhs, αht),
∂Jvh = αhJX ˜ H = O(αh) for h → 0,
π ◦ vh(0, t) = (ch(αht), ch(1 − αht)) for 0 ≤ t ≤ 1/(2αh), (203)
vh(s, 0) ∈ N ∗ ∆M×M for 0 ≤ s ≤ 1, (204)
vh(s, 0) ∈ N ∗ ∆ Θ M for s ≥ 1, (205)
vh(s, 1/(2αh)) ∈ N ∗ ∆M×M for s ≥ 0, (206)
lim
s→+∞ vh(s, t) = (x(αht), −x(1 − αht)). (207)
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