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

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 MCP(K,N)-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 CD(K,N) and a dimensional Bochner inequality on signed distance functions. Moreover we obtain a measure-theoretic splitting theorem for infinitesimally Hilbertian, essentially nonbranching spaces satisfying MCP(0,N).

Ricci curvature, optimal transport, Laplacian comparison, distance function
Mathematical Subject Classification 2010
Primary: 49J52, 53C23
Received: 20 November 2018
Revised: 21 July 2019
Accepted: 6 September 2019
Published: 10 November 2020
Fabio Cavalletti
Mathematics Area
Andrea Mondino
Mathematical Institute
University of Oxford
United Kingdom
Mathematics Institute
University of Warwick
United Kingdom