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The energy of a solution u ∈ M K α (γ1, γ2; x) is uniformly bounded:
�
E(u) :=
|∂su| 2 ds dt ≤ËL1(γ1) +ËL2(γ2) −�H1⊕H2(x). (192)
]0,+∞[×]0,1[
Let α0 > 0. For a generic choice of g1, g2, J1, and J2, both M K 0 (γ1, γ2; x) and M K α0 (γ1, γ2; x) are
smooth oriented manifolds of dimension
m Λ (γ1, L1) + m Λ (γ2, L2) − µ Θ (x) − n,
for every γ1 ∈ P Λ (L1), γ2 ∈ P Λ (L2), x ∈ P Θ (H1 ⊕ H2) (see Proposition 4.5). The usual
counting process defines the chain maps
K Λ 0 , K Λ α0 : (M(ËΛ L1 , g1) ⊗ M(ËΛ L2 , g2))∗ −→ F Θ ∗−n(H1 ⊕ H2, J1 ⊕ J2),
and we wish to prove that KΛ 0 ⊗ KΛ α0 is chain homotopic to KΛ α0 ⊗ KΛ 0 . Since KΛ is homotopic
α0
to KΛ α1 for α0, α1 ∈]0, +∞[, we may as well assume that α0 is small. Moreover, since the chain
maps KΛ 0 and KΛ α0 preserve the filtrations of the Morse and Floer complexes given by the action
sublevels
ËΛ
(γ1) +ËΛ
(γ2) ≤ A,�H1⊕H2(x) ≤ A,
L1 L2
we can work with the subcomplexes corresponding to a fixed (but arbitrary) action bound A. We
also choose the Lagrangians L1 and L2 to be non-negative, so that every orbit has non-negative
action.
Convergence. Fix some γ1 ∈ P(L1), γ2 ∈ P(L2), and x ∈ P Θ (H1 ⊕ H2), such that
m Λ (γ1) + m Λ (γ2) − µ Θ (x) = n. (193)
Let (αh) be an infinitesimal sequence of positive numbers, let uh be an element of M K αh (γ1, γ2; x),
and let ch be the projection onto M × M of the closed curve u(0, ·). By (188), ch is an element of
W u (γ1; −grad g1ËΛ L1 ) × W u (γ2; −grad g2ËΛ L2 ). The latter space is pre-compact in W 1,2 ([0, 1], M ×
M). By the argument of breaking gradient flow lines, up to a subsequence we may assume that
(ch) converges in W 1,2 to a curve c in W u (˜γ1; −grad g1ËΛ L1 ) × W u (˜γ2; −grad g2ËΛ L2 ), for some
˜γ1 ∈ P(L1) and ˜γ2 ∈ P(L1) such that
either m Λ (˜γ1) + m Λ (˜γ2) < m Λ (γ1) + m Λ (γ2) or (˜γ1, ˜γ2) = (γ1, γ2). (194)
Similarly, the upper bound (192) on the energy E(uh) implies that (uh) converges in C ∞ loc on
[0, +∞[×[0, 1] \ {(0, 0), (0, 1)}, using the standard argument excluding bubbling off of spheres and
disks. In particular,
and
The limit u satisfies equation (187), and
with ˜x in P Θ (H1 ⊕ H2) such that
(u(s, 0), −u(s, 1)) ∈ N ∗ ∆ Θ M, ∀s > 0, (195)
π ◦ u(0, t) = c(t), ∀t ∈]0, 1[. (196)
lim u(s, ·) = ˜x,
s→+∞
either µ Θ (˜x) > µ Θ (x) or ˜x = x, (197)
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