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New formulas for the
Laplacian of distance functions and applications
Fabio Cavalletti and Andrea Mondino |
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Vol. 13 (2020), No. 7, 2091–2147
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Abstract |
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The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary -Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense (more precisely in essentially nonbranching -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 and a dimensional Bochner inequality on signed distance functions. Moreover we obtain a measure-theoretic splitting theorem for infinitesimally Hilbertian, essentially nonbranching spaces satisfying . |
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Keywords
Ricci curvature, optimal transport, Laplacian comparison,
distance function
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Mathematical Subject Classification 2010
Primary: 49J52, 53C23
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Milestones
Received: 20 November 2018
Revised: 21 July 2019
Accepted: 6 September 2019
Published: 10 November 2020
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Authors |
| Fabio Cavalletti | |
| Mathematics Area SISSA Trieste Italy |
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| Andrea Mondino | |
| Mathematical Institute University of Oxford Oxford United Kingdom |
Mathematics Institute University of Warwick Coventry United Kingdom |
