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

Université de Nice-Sophia Antipolis, Laboratoire J.-A. Dieudonné, Parc Valrose, 06108 Nice Cedex 02, France. castelpietra@unice.fr; rifford@unice.fr

Received: 6 December 2008
Revised: 31 March 2009

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 [Trans. Amer. Math. Soc. 353 (2001) 21–40], is due to Li and Nirenberg [Comm. Pure Appl. Math. 58 (2005) 85–146]. 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 C⁴-deformation of the round sphere, then all its tangent nonfocal domains are strictly uniformly convex.