Methods of infinite dimensional Morse theory for geodesics on 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.

81 pages, 1 figure. v7: Published version

References in corpus (4)

• On the energy functional on Finsler manifolds and applications to stationary spacetimes
• Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric
• The index of a geodesic in a Randers space and some remarks about the lack of regularity of the energy functional of a Finsler metric
• The splitting lemmas for nonsmooth functionals on Hilbert spaces I

Cited by in corpus (2)

• The splitting lemmas for nonsmooth functionals on Hilbert spaces I
• Splitting lemmas for the Finsler energy functional on the space of $H^1$-curves