New formulas for the Laplacian of distance functions and applications

TL;DR

This paper derives an exact formula for the Laplacian of distance functions in metric measure spaces with Ricci curvature bounds, revealing classical bounds, singular parts, and applications to Bochner inequalities and splitting theorems.

Contribution

It introduces a new representation formula for the Laplacian of 1-Lipschitz functions in synthetic Ricci curvature spaces, applicable to Riemannian manifolds and MCP spaces.

Findings

Exact Laplacian representation formula in MCP(K,N)-spaces.

Equivalence of CD(K,N) and dimensional Bochner inequality.

Measure-theoretic splitting theorem for MCP(0,N) spaces.

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 non-branching MCP(K,N)-spaces). Such a representation formula makes apparent the classical upper bounds and also some new lower bounds, together with a precise description of the singular part. The exact representation formula for the Laplacian of 1-Lipschitz functions (in particular for distance functions) holds also (and seems new) in a general complete Riemannian manifold. We apply these results to prove the equivalence of CD(K,N) and a dimensional Bochner inequality on signed distance functions. Moreover we obtain a measure-theoretic Splitting Theorem for infinitesimally Hilbertian essentially non-branching spaces verifying MCP(0,N).