Minimizing Closed Geodesics via Critical Points of the Uniform Energy

TL;DR

This paper explores the properties of 1/k-geodesics and their connection to critical points of the uniform energy, using energy methods and convergence analysis to deepen understanding of geodesic minimization and critical point theory.

Contribution

It introduces the concept of balanced points of the uniform energy and relates them to 1/k-geodesics, extending the analysis to Gromov-Hausdorff limits and connecting half-geodesics with critical points of the distance function.

Findings

Balanced points of the uniform energy persist under Gromov-Hausdorff convergence.

Half-geodesics are related to Grove-Shiohama critical points.

Complete characterization of the differentiability of the distance function.

Abstract

In this paper we study 1/k-geodesics, those closed geodesics that minimize on any subinterval of length $l(\gamma)/k$. We employ energy methods to provide a relationship between the 1/k-geodesics and what we define as the balanced points of the uniform energy. We show that classes of balanced points of the uniform energy persist under the Gromov-Hausdorff convergence of Riemannian manifolds. Additionally, we relate half-geodesics (1/2-geodesics) to the Grove-Shiohama critical points of the distance function. This relationship affords us the ability to study the behavior of half-geodesics via the well developed field of critical point theory. Along the way we provide a complete characterization of the differentiability of the Riemannian distance function.