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

TL;DR

This paper proves that on a smooth Riemannian manifold, the Laplacian of the distance function to a point becomes negatively infinite at the cut locus, providing insight into the behavior of the Laplacian near singularities.

Contribution

It establishes that the Laplacian of the distance function is negatively infinite at the cut locus in the barrier sense on smooth Riemannian manifolds.

Findings

Laplacian of the distance function is -∞ at the cut locus

Provides a barrier sense interpretation of the Laplacian behavior

Enhances understanding of geometric analysis on Riemannian manifolds

Abstract

We show that, on a smooth riemannian manifold, the laplacian of the distance function to a point $b$ is $-\infty$ in the sense of barriers, at every point of the cut locus with respect to $b$.