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commute. Our first aim in this section is to show that the lower two triangles commute. We
actually work at the chain level, showing that the triangles
Mc
Mj(f, gM) ��
Mj(ËΛ
�
L , g
����
���
C �����
Λ )
��
ΦΛ Mev ��
Mj(f, gM)
����
���
L ���
Ev ���
(H, J)
F Λ j
commute up to chain homotopies.
Indeed, a homotopy P C between C and ΦΛ L ◦ Mc is defined by the following spaces: given
x ∈ crit(f) and y ∈ PΛ (H), set
M C �
P (x, y) := (α, u) ∈]0, +∞[×C ∞ ([0, +∞[×Ì, T ∗ �
�
M) � ∂J,H(u) = 0,
φ Λ −α(π ◦ u(0, ·)) ≡ q ∈ W u �
(x) ,
where φΛ is the flow of −gradËΛ
L on Λ1 (M).
Similarly, the definition of the homotopy P Ev between Ev ◦ ΦΛ L and Mev is obtained from the
composition of 3 homotopies based on the following spaces: Given γ ∈ PΛ (L) and x ∈ crit(f),
set
M Ev
� �
�
P1 (γ, x) := (α, u) �α ∈ [1, +∞[, u ∈ C ∞ ([0, α] ×Ì, T ∗ M)
solves ∂J,H(u) = 0, u(α, t) ∈ÇM ∀t ∈Ì,
u(α, 0) ∈ W s (x),
π ◦ u(0, ·) ∈ W u �
(γ; −gradËΛ
L ) ,
M Ev
� �
�
P2 (γ, x) := (α, u) �α ∈ [0, 1], u ∈ C ∞ ([0, 1] ×Ì, T ∗ M)
solves ∂J,H(u) = 0, u(1, t) ∈ÇM ∀t ∈Ì,
u(α, 0) ∈ W s (x),
π ◦ u(0, ·) ∈ W u �
(γ; −gradËΛ
L ) , and
M Ev
� �
�
P3 (γ, x) := (α, u) �α ∈]0, 1], u ∈ C ∞ ([0, α] ×Ì, T ∗ M)
solves ∂J,H(u) = 0, u(α, t) ∈ÇM ∀t ∈Ì,
u(0, 0) ∈ W s (x),
π ◦ u(0, ·) ∈ W u �
(γ; −gradËΛ
L) .
Moreover, recalling the following space on which Mev is based,
we make the following observation.
MMev(γ, x) = W u (γ; −gradËΛ L ) ∩ ev −1� W s (x; −gradf) � ,
4.9. Proposition. Given the finite set MMev(γ, x), there exists αo > 0 such that for each c ∈
MMev(γ, x) and α ∈ (0, αo] the problem
u ∈ [0, α] ×Ì→T ∗ M, ∂J,Hu = 0,
u(α, t) ∈ÇM ∀t ∈Ì, π ◦ u(0, ·) = c,
has a unique solution with the same coherent orientation as c.
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