PDF (1016 kB)

Since the inclusion c : M ֒→ Λ 1 (M) induces an injective homomorphism between the singular
homology groups, the image c∗([M]) of the fundamental class of the oriented manifold M does not
vanish in Λ 1 (M). By (ii) and (v), the actionËL2 of every constant curve in M does not exceed 0.
So we can regard c∗([M]) as a non vanishing element of the homology of the sublevel {ËL2 ≤ 0}.
The singular homology of {ËL2 ≤ 0} is isomorphic to the homology of the subcomplex of the
Morse complex M∗(ËΛ L2 , g2) generated by the critical points ofËΛ L2 whose action does not exceed
0. By (65), these critical points are the equilibrium solutions q, with q ∈ critV2. By (ii), (iii), and
(64), the only critical point of index n in this sublevel is q0. It follows that the Morse homological
counterpart of c∗([M]) is ±q0. In particular, q0 ∈ Mn(ËL2) is a cycle. SinceËΛ L2 (q0) = 0,
ËΛ L2 (γ) ≤ 0, ∀γ ∈ W u (q0, −gradËΛ L2 ). (66)
We now regard the pair (q0, q0) as an element of P Θ (L1 ⊕ L2). We claim that if ǫ is small
enough, (iv) implies that (q0, q0) is a non-degenerate minimizer forËL1⊕L2 on the space of figure-8
loops Θ 1 (M). The second differential ofËΘ L1⊕L2 at (q0, q0) is the quadratic form
d 2ËΘ L1⊕L2 (q0, q0)[(ξ1, ξ2)] 2 =
� 1
0
�
〈ξ ′ 1 , ξ′ 1 〉 − 〈Hess V1(t, q0)ξ1, ξ1〉 + 〈ξ ′ 2 , ξ′ 2 〉 − 〈Hess V2(q0)ξ2,
�
ξ2〉 dt,
on the space of curves (ξ1, ξ2) in the Sobolev space W 1,2 (]0, 1[, Tq0M × Tq0M) satisfying the
boundary conditions
By (59) we can find α > 0 such that that
ξ1(0) = ξ1(1) = ξ2(0) = ξ2(1).
Hess V1(t, q0) ≤ −αI.
By comparison, it is enough to show that the quadratic form
Qǫ(u1, u2) :=
� 1
0
(u ′ 1(t) 2 + αu1(t) 2 + u ′ 2(t) 2 − ǫu2(t) 2 )dt
is coercive on the space
� (u1, u2) ∈ W 1,2 (]0, 1[,Ê2 ) | u1(0) = u1(1) = u2(0) = u2(1) � .
When ǫ = 0, the quadratic form Q0 is non-negative. An isotropic element (u1, u2) for Q0 would
solve the boundary value problem
−u ′′
1(t) + αu1(t) = 0, (67)
(t) = 0, (68)
−u ′′
2
u1(0) = u1(1) = u2(0) = u2(1), (69)
u ′ 1 (1) − u′ 1 (0) = u′ 2 (0) − u′ 2 (1). (70)
By (68) u2 is constant, so by (69) and (70) u1 is a periodic solution of (67). Since α is positive, u1
is zero and by (69) so is u2. Since the bounded self-adjoint operator associated to Q0 is Fredholm,
we deduce that Q0 is coercive. By continuity, Qǫ remains coercive for ǫ small. This proves our
claim.
Let H1 and H2 be the Hamiltonians which are Fenchel dual to L1 and L2. In order to simplify
the notation, let us denote by (q0, q0) also the constant curve in T ∗ M 2 identically equal to
((q0, 0), (q0, 0)). Then (q0, q0) is a non-degenerate element of P Θ (H1 ⊕ H2), and it has Maslov
index
µ Θ (q0, q0) = 0.
48