Estimates and computations in Rabinowitz-Floer homology

TL;DR

This paper constructs a chain-level exact sequence in Rabinowitz-Floer homology for Liouville domains and generalizes it to broader classes of Hamiltonians, providing new insights into the algebraic and geometric structures involved.

Contribution

It presents a chain level construction of the Rabinowitz-Floer homology exact sequence and extends it to more general convex subsets of cotangent bundles.

Findings

Chain complex sequence induces Rabinowitz-Floer homology sequence.

Extension to convex subsets of cotangent bundles with Lagrangian graphs.

Uniform estimates for solutions with quadratic Hamiltonian growth.

Abstract

The Rabinowitz-Floer homology of a Liouville domain W is the Floer homology of the free period Hamiltonian action functional associated to a Hamiltonian whose zero energy level is the boundary of W. It has been introduced by K. Cieliebak and U. Frauenfelder. Together with A. Oancea, the same authors have recently computed the Rabinowitz-Floer homology of the cotangent disk bundle D^*M of a closed manifold M, by establishing a long exact sequence. The first aim of this paper is to present a chain level construction of this exact sequence. In fact, we show that this sequence is the long homology sequence induced by a short exact sequence of chain complexes, which involves the Morse chain complex and the Morse differential complex of the energy functional for closed geodesics on M. These chain maps are defined by considering spaces of solutions of the Rabinowitz-Floer equation on half-cylinders, with suitable boundary conditions which couple them with the negative gradient flow of the geodesic energy functional. The second aim is to generalize this construction to the case of a fiberwise uniformly convex compact subset W of T^*M whose interior part contains a Lagrangian graph. Equivalently, W is the energy sublevel associated to an arbitrary Tonelli Lagrangian L on TM and to any energy level which is larger than the strict Mane' critical value of L. In this case, the energy functional for closed geodesics is replaced by the free period Lagrangian action functional associated to a suitable calibration of L. An important issue in our analysis is to extend the uniform estimates for the solutions of the Rabinowitz-Floer equation to Hamiltonians which have quadratic growth in the momenta. These uniform estimates are obtained by suitable versions of the Aleksandrov maximum principle.