Laplacian of the distance function on the cut locus on a Riemannian manifold

Abstract

In this note, we prove that, on a Riemannian manifold M, the Laplacian of the distance function to a point b is −∞ in the sense of barriers, at every point of the cut locus of M with respect to b. We apply this result to an obstacle-type variational problem where the obstacle is the distance function. It allows us to replace the latter with a smoother obstacle.

Cut Locus on Compact Manifolds and Uniform Semiconcavity Estimates for a Variational Inequality

“…(Regularization of the obstacle,[16]) For any m > 0, there exists a function d b which is smooth on M \ {b}, such thatu d m d b d b on M, and d b < d b on Cut b (M).In particular, u d m is also the solution of the obstacle problemmin M mu : u ∈ H 1 (M), u d b . (3.1)One could adapt to the manifold framework the regularity theorems for the classical obstacle problem on a euclidean domain and, with the preceding lemma, deduce the regularity of u d m .…”