Some obstructions to contraction theorems on the half-sphere

2024/02/07 by Max Fathi, Fathi, Max, Matthieu Fradelizi +5 · 2 citations
Mathematics · #advanced mathematical theories #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2402.04649

Abstract

Caffarelli's contraction theorem states that probability measures with uniformly logconcave densities on R d can be realized as the image of a standard Gaussian measure by a globally Lipschitz transport map. We discuss some counterexamples and obstructions that prevent a similar result from holding on the half-sphere endowed with a uniform measure, answering a question of Beck and Jerison.

Cited by

• A Generalization of Caffarelli's Contraction Theorem to Nearly Spherical Manifolds
• Contractive transport maps from \mathbbS² to nearly spherical surfaces with positive Ricci curvature

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