Morse theory for the uniform energy
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.
Keywords
Cite
@article{arxiv.1609.09357,
title = {Morse theory for the uniform energy},
author = {Ian Adelstein and Jonathan Epstein},
journal= {arXiv preprint arXiv:1609.09357},
year = {2017}
}
Comments
This revised version includes a proof of Lemma 3.2 and a more general procedure for restarting the negative gradient flow of the Morse energy