Displacement energy of unit disk cotangent bundles

TL;DR

This paper establishes an upper bound for the displacement energy of unit disk cotangent bundles over open Riemannian manifolds, linking it to the manifold's inner radius and exploring implications for symplectic embeddings and billiard trajectories.

Contribution

It provides a new upper bound for displacement energy in cotangent bundles of open manifolds, connecting geometric and symplectic properties.

Findings

Displacement energy is bounded above by a constant times the inner radius of the manifold.

Results imply existence of short periodic billiard trajectories.

Results contribute to understanding symplectic embedding problems.

Abstract

We give an upper bound of a Hamiltonian displacement energy of a unit disk cotangent bundle $D^*M$ in a cotangent bundle $T^*M$, when the base manifold $M$ is an open Riemannian manifold. Our main result is that the displacement energy is not greater than $C r(M)$, where $r(M)$ is the inner radius of $M$, and $C$ is a dimensional constant. As an immediate application, we study symplectic embedding problems of unit disk cotangent bundles. Moreover, combined with results in symplectic geometry, our main result shows the existence of short periodic billiard trajectories and short geodesic loops.