New formulas for the Laplacian of distance functions and applications

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).

arXiv commentFinal version to appear in Analysis and PDE

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• Maximal volume entropy rigidity for ${\mathsf{RCD}}^{\ast }(-(N-1),N)$ spaces
• Sharp log-Sobolev inequalities in $\mathsf{CD}(0,N)$ spaces with applications
• Topological regularity of isoperimetric sets in PI spaces having a deformation property
• Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary
• On the notion of Laplacian bounds on $\mathrm{RCD}$ spaces and applications
• Hölder continuity of tangent cones in RCD(K,N) spaces and applications to non-branching
• Displacement convexity of Entropy and the distance cost Optimal Transportation
• Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures