PDF (1016 kB)

(L0) Ω every solution γ ∈ P Ω (L) is non-degenerate, meaning that there are no non-zero Jacobi
vector fields along γ which vanish for t = 0 and for t = 1.
(L0) Λ every solution γ ∈ P Λ (L) is non-degenerate, meaning that there are no non-zero periodic
Jacobi vector fields along γ.
These conditions hold for a generic choice of L, in several reasonable senses. Notice that (L0) Λ
forces L to be explicitly time-dependent, otherwise γ ′ would be a non-zero 1-periodic Jacobi vector
field along γ, for every non-constant periodic solution γ. In the fixed ends case instead, L is allowed
to be autonomous.
We need also to consider the following Lagrangian problem for figure-8 loops. Let L1, L2 ∈
C ∞ ([0, 1] ×TM) be Lagrangians satisfying (L1) and (L2). Let P Θ (L1 ⊕L2) be the set of all pairs
(γ1, γ2), with γj : [0, 1] → M solution of the Lagrangian system given by Lj, such that
γ1(0) = γ1(1) = γ2(0) = γ2(1),
1�
i=0 j=1
2�
(−1) i ∂vLj(i, γj(i), γ ′ j(i)) = 0. (19)
The elements of P Θ (L1⊕L2) are precisely the critical points of the functionalËL1⊕L2 =ËL1 ⊕ËL2
restricted to the submanifold Θ 1 (M),
ËL1⊕L2(γ1, γ2) =
� 1
0
L1(t, γ1(t), γ ′ 1 (t))dt +
� 1
0
L2(t, γ2(t), γ ′ 2 (t))dt.
Such a restriction is denoted byËΘ L1⊕L2 . If we denote by m Θ (γ1, γ2) the Morse index of (γ1, γ2) ∈
P Θ (L1 ⊕ L2), we clearly have
max{m Ω (γ1; L1) + m Ω (γ2; L2), m Λ (γ1; L1) + m Λ (γ2; L2) − n} ≤ m Θ (γ1, γ2)
≤ min{m Ω (γ1; L1) + m Ω (γ2; L2) + n, m Λ (γ1; L1) + m Λ (γ2; L2)}.
The non-degeneracy of every critical point ofËΘ L1⊕L2 is equivalent to the condition:
(L0) Θ every solution (γ1, γ2) ∈ P Θ (L1 ⊕ L2) is non-degenerate, meaning that there are no nonzero
pairs of Jacobi vector fields ξ1, ξ2 along γ1, γ2 such that
1�
i=0 j=1
ξ1(0) = ξ1(1) = ξ2(0) = ξ2(1),
2� �
∂qvLj(i, γj(i), γ ′ j(i))ξj(i) + ∂vvLj(i, γj(i), γ ′ j(i))ξ ′ j(i) � = 0.
This condition allows both L1 and L2 to be autonomous. It also allows L1 = L2, but this excludes
the autonomous case (otherwise pairs (γ, γ) with γ ∈ PΛ (L1) = PΛ (L2) non-constant would
violate (L0) Θ ).
Assumption (L1) implies that L is bounded below, and it can be shown that (L1) and (L2)
imply thatËL satisfies the Palais-Smale condition on Ω1 (M, q0) and on Λ1 (M), with respect to
the standard W 1,2-metric 〈〈ξ, η〉〉γ :=
� 1
0
� �
〈ξ(t), η(t)〉γ(t) + 〈∇tξ(t), ∇tη(t)〉 γ(t) dt,
where 〈·, ·〉 is a metric on M, and ∇t is the corresponding Levi-Civita covariant derivation along
γ. The same is true for the functionalËL1⊕L2 on Θ 1 (M). See for instance the appendix in [AF07]
for a proof of the Palais-Smale condition under general non-local conormal boundary conditions,
including all the case treated here. We conclude that under the assumptions (L0) Ω (resp. (L0) Λ ,
resp. (L0) Θ ), (L1), (L2), the functionËΩ L (resp.ËΛ L , resp.ËΘ L1⊕L2 ) belongs to F(Ω 1 (M, q0), 〈〈·, ·〉〉)
(resp. F(Λ 1 (M), 〈〈·, ·〉〉), resp. F(Θ 1 (M), 〈〈·, ·〉〉)).
18