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New formulas for the Laplacian of distance functions and applications

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arxiv 1803.09687 v2 pith:DZDHBPEO submitted 2018-03-26 math.MG math.FA

Fabio Cavalletti , Andrea Mondino This is my paper · ORCID
classification math.MGmath.FA
keywords distancefunctionsboundsformulalaplacianrepresentationspacesessentially
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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 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).

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Quasi Curvature-Dimension Condition with applications to sub-Riemannian manifolds

    math.FA 2019-08 accept novelty 8.0 of 10

    A quasi-convex relaxation of the curvature-dimension condition gives dimension-independent Poincaré and log-Sobolev constants on Heisenberg groups and other sub-Riemannian manifolds, up to a universal factor.

  2. The Heintze-Karcher inequality for metric measure spaces

    math.DG 2019-08 accept novelty 7.0 of 10

    The Heintze-Karcher inequality, a Riemannian volume comparison theorem, is generalized to essentially non-branching metric measure spaces with lower Ricci curvature bounds, with a rigidity result for RCD spaces.

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