#### Vol. 13, No. 7, 2020

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New formulas for the Laplacian of distance functions and applications

### Fabio Cavalletti and Andrea Mondino

Vol. 13 (2020), No. 7, 2091–2147
##### 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 $\mathsf{MCP}\left(0,N\right)$.

##### Keywords
Ricci curvature, optimal transport, Laplacian comparison, distance function
##### Mathematical Subject Classification 2010
Primary: 49J52, 53C23