New formulas for the Laplacian of distance functions and applications

Abstract

The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary $1$-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense (more precisely in essentially nonbranching $\mathsf{MCP}\left(K,N\right)$-spaces). Such a representation formula makes apparent the classical upper bounds together with lower bounds and a precise description of the singular part. The exact representation formula for the Laplacian of a general 1-Lipschitz function holds also (and seems new) in a general complete Riemannian manifold.

We apply these results to prove the equivalence of $\mathsf{CD}\left(K,N\right)$ and a dimensional Bochner inequality on signed distance functions. Moreover we obtain a measure-theoretic splitting theorem for infinitesimally Hilbertian, essentially nonbranching spaces satisfying $\mathsf{MCP}\left(0,N\right)$.