Minimizing Closed Geodesics via Critical Points of the Uniform Energy

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.

arXiv commentThis second version includes a new section (now Section 5) which studies the uniform energy functional under Gromov-Hausdorff convergence