A Generalization of Caffarelli's Contraction Theorem to Nearly Spherical Manifolds

2025/12/01 by Ge, Yuxin, Serres, Jordan · 1 citation
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Control and Stability of Dynamical Systems #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Probability (math.PR)

paper · doi:10.48550/arxiv.2512.01496

openalex publication_date 2025/12/01 · openalex created_date 2025/12/03 · openalex updated_date 2026/07/28

Abstract

We show that every nearly spherical manifold can be realized as the volume-preserving image of a round sphere, via the Brenier-McCann optimal transport map. This theorem extends Caffarelli's contraction theorem to nearly spherical manifolds and yields, as a corollary, a proof of a perturbative form of Milman's conjecture. The proof is based on a novel stability result for optimal transport maps on the sphere.

Citations

• Stability of the Kim--Milman flow map
• Unstable optimal transport maps
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• On the second boundary value problem for Monge-Ampere type equations and optimal transportation
• On the regularity of the polar factorization for time dependent maps
• Polar factorization and monotone rearrangement of vector‐valued functions

Cited by

• Stability of the Kim--Milman flow map
• Contraction Maps Generated by Inverse Mean Curvature Flow

Related

• Contractive transport maps from \mathbbS² to nearly spherical surfaces with positive Ricci curvature
• Some obstructions to contraction theorems on the half-sphere
• Contraction Maps Generated by Inverse Mean Curvature Flow
• Regularity of optimal transport maps on multiple products of spheres
• Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular)