The Palais–Smale condition for the Hamiltonian action on a mixed regularity space of loops in cotangent bundles and applications

Abstract

We show that the Hamiltonian action satisfies the Palais-Smale condition over a "mixed regularity" space of loops in cotangent bundles, namely the space of loops with regularity H s , s ∈ (1 2 , 1), in the base and H 1−s in the fiber direction. As an application, we give a simplified proof of a theorem of Hofer-Viterbo on the existence of closed characteristic leaves for certain contact type hypersufaces in cotangent bundles. Mathematics Subject Classification 37J45 One of the central problem in the theory of Hamiltonian systems is to find (one-)periodic solutions of (1.1). Such periodic solutions can be found as critical points of a suitable action Communicated by P. Rabinowitz.

Morse homology for the Hamiltonian action in cotangent bundles

“…Referring to [10] for the details, we see that, for s > 1 2 , the fractional Sobolev space H s (T, M ), T := R/Z, of H s -loops in M has a natural structure of Hilbert manifold, and for any r ∈ [−s, s] (see also Lemma 2.5) there exists a vector bundle…”

“…In the theorem below we summarize the properties of the functional A H , referring to [10, Section 2] for the proof. We shall notice that the Palais-Smale condition is proved in [10] only for Hamiltonians which are kinetic (up to a constant) outside a compact set, that is only for θ ≡ 0 and U ≡ c, however the proof extends verbatim to the case in which θ is an arbitrary one-form on M and U is a time-depending potential.…”

“…We start observing that the metric on M 1−s defined in (2.3) is, on every bounded set of M 1−s , equivalent to the metric induced by embedding M isometrically in R N , see Lemma 2.5 in [10]. Even though the two metrics might fail to be globally equivalent (see Appendix B of [10]), the local equivalence is enough to apply Proposition 4.1 on every bounded set of M 1−s , and actually on every set of the form π −1 1−s (A), A ⊂ H s (T, M ) bounded. Thus, we consider a sequence {ϕ * ,ℓ } ℓ∈N of local parametrizations such that…”

“…In this paper we upgrade the results in [10] by showing that one can obtain a genuine Morse complex for A H using the Morse complex approach which is developed in [1,2,3,5]. In this approach, one constructs a chain complex by looking at the one-dimensional intersections of unstable and stable manifolds of pairs of critical points.…”

“…Despite the great success of the theory of Floer, the question remained how far the original approach by Rabinowitz, Conley, and Zehnder can be generalized to manifolds different from the torus. In [10] we started addressing this question by showing that, for a Hamiltonian H : T × T * M → R with quadratic growth at infinity, the Hamiltonian action A H : M 1−s → R, s ∈ (1/2, 1), (1.2) satisfies the Palais-Smale condition, M 1−s being the Hilbert bundle over the Hilbert manifold of loops H s (T, M ), s ∈ (1/2, 1), whose typical fibre is given by the space of H 1−s -sections of the pull-back bundle c * (T * M ), where c : T → M is any smooth loop. Roughly speaking, instead of considering loops with Sobolev regularity 1/2, one considers loops in T * M whose projection to the base has regularity larger than 1/2 (but strictly less than 1), thus making them continuous in the base direction, and whose projection to the fiber has regularity smaller than 1/2 (but strictly larger than L 2 -regularity) in such a way that the mean regularity is 1/2.…”