Regularity properties of the distance function to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and application in Riemannian geometry

TL;DR

This paper investigates the regularity of the distance function to conjugate and cut loci in the context of viscosity solutions of Hamilton-Jacobi equations, with applications to Riemannian geometry, demonstrating local semiconcavity and Lipschitz properties.

Contribution

It proves local semiconcavity of the conjugate locus distance function and Lipschitz continuity of the cut locus distance function for viscosity solutions, extending previous results and applying to geometric settings.

Findings

Distance to conjugate locus is locally semiconcave.

Distance to cut locus is locally Lipschitz.

Riemannian manifolds close to spheres have strictly convex tangent nonfocal domains.

Abstract

Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka, is due to Li and Nirenberg. Finally, we give applications of our results in Riemannian geometry. Namely, we show that the distance function to the conjugate locus on a Riemannian manifold is locally semiconcave. Then, we show that if a Riemannian manifold is a deformation of the round sphere, then all its tangent nonfocal domains are strictly uniformly convex.