Hofer geometry and cotangent fibers

TL;DR

This paper demonstrates that for certain Riemannian manifolds, the Hamiltonian diffeomorphism group of the cotangent bundle contains large, bi-Lipschitz embedded subgroups with respect to Hofer's metric, generated by reparametrizations of geodesic flows.

Contribution

It introduces a method to embed infinite-dimensional normed vector spaces into the Hamiltonian diffeomorphism group of cotangent bundles for specific manifolds, using Floer homology techniques.

Findings

Existence of bi-Lipschitz embedded infinite-dimensional subgroups

Subgroups supported in neighborhoods of the zero section

Orbit of a cotangent fiber is quasi-isometrically embedded

Abstract

For a class of Riemannian manifolds that include products of arbitrary compact manifolds with manifolds of nonpositive sectional curvature on the one hand, or with certain positive-curvature examples such as spheres of dimension at least 3 and compact semisimple Lie groups on the other, we show that the Hamiltonian diffeomorphism group of the cotangent bundle contains as subgroups infinite-dimensional normed vector spaces that are bi-Lipschitz embedded with respect to Hofer's metric; moreover these subgroups can be taken to consist of diffeomorphisms supported in an arbitrary neighborhood of the zero section. In fact, the orbit of a fiber of the cotangent bundle with respect to any of these subgroups is quasi-isometrically embedded with respect to the induced Hofer metric on the orbit of the fiber under the whole group. The diffeomorphisms in these subgroups are obtained from reparametrizations of the geodesic flow. Our proofs involve a study of the Hamiltonian-perturbed Floer complex of a pair of cotangent fibers (or, more generally, of a conormal bundle together with a cotangent fiber). Although the homology of this complex vanishes, an analysis of its boundary depth yields the lower bounds on the Lagrangian Hofer metric required for our main results.