Minimizing closed geodesics via critical points of the uniform energy

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 halfgeodesics 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.

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Accepted 23 March 2015

Published 16 September 2016