Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds

TL;DR

This paper develops advanced Morse theory techniques for analyzing geodesics on Finsler manifolds, extending classical results and providing new tools for studying critical points of energy functionals in infinite-dimensional settings.

Contribution

It introduces novel shifting theorems and splitting lemmas for Finsler energy functionals, and extends results on closed geodesics from Riemannian to Finsler manifolds.

Findings

Proved shifting theorems for critical groups of Finsler energy functionals.

Established splitting lemmas for functionals on Banach manifolds of curves.

Extended existence results of infinitely many closed geodesics to Finsler manifolds.

Abstract

We prove the shifting theorems of the critical groups of critical points and critical orbits for the energy functionals of Finsler metrics on Hilbert manifolds of $H^1$-curves, and two splitting lemmas for the functionals on Banach manifolds of $C^1$-curves. Two results on critical groups of iterated closed geodesics are also proved; their corresponding versions on Riemannian manifolds are based on the usual splitting lemma by Gromoll and Meyer (1969). Our approach consists in deforming the square of the Finsler metric in a Lagrangian which is smooth also on the zero section and then in using the splitting lemma for nonsmooth functionals that the author recently developed in Lu (2011, 0000, 2013). The argument does not involve finite-dimensional approximations and any Palais' result in Palais (1966). As an application, we extend to Finsler manifolds a result by V. Bangert and W. Klingenberg (1983) about the existence of infinitely many, geometrically distinct, closed geodesics on a compact Riemannian manifold.