Splitting lemmas for the Finsler energy functional on the space of $H^1$-curves

Abstract

We establish the splitting lemmas (or generalized Morse lemmas) for the energy functionals of Finsler metrics on the natural Hilbert manifolds of $H^1$-curves around a critical point or a critical ${\mathbb{R}}^1$ orbit of a Finsler isometry invariant closed geodesic. They are the desired generalization on Finsler manifolds of the corresponding Gromoll-Meyer's splitting lemmas on Riemannian manifolds (\cite{GM1, GM2}). As an application we extend to Finsler manifolds a result by Grove and Tanaka \cite{GroTa78, Tan82} about the existence of infinitely many, geometrically distinct, isometry invariant closed geodesics on a closed Riemannian manifold.

56 pages, Latex, latest version, to appear in Proceedings of the London Mathematical Society

References in corpus (2)

• On the multiplicity of isometry-invariant geodesics on product manifolds
• Isometry-invariant geodesics and the fundamental group

Cited by in corpus (1)

• Parameterized splitting theorems and bifurcations for potential operators