Morse theory for the uniform energy

TL;DR

This paper develops a Morse theory framework for the uniform energy on manifolds, utilizing directional derivatives of the distance function to analyze and improve properties of closed geodesics.

Contribution

It introduces a novel Morse theory approach for the uniform energy using directional derivatives, enabling better analysis of geodesic minimization.

Findings

Reformulation of Morse theory for the uniform energy

Application to flat torus showing improved geodesic properties

New method to restart gradient flow at critical points

Abstract

In this paper we develop a Morse theory for the uniform energy. We use the one-sided directional derivative of the distance function to study the minimizing properties of variations through closed geodesics. This derivative is then used to define a one-sided directional derivative for the uniform energy which allows us to identify gradient-like vectors at those points where the function is not differentiable. These vectors are used to restart the standard negative gradient flow of the Morse energy at its critical points. We illustrate this procedure on the flat torus and demonstrate that the restarted flow improves the minimizing properties of the associated closed geodesics.