A Generalization of Caffarelli's Contraction Theorem to Nearly Spherical Manifolds
Analysis of PDEs
2025-12-02 v1 Differential Geometry
Functional Analysis
Probability
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.
Keywords
Cite
@article{arxiv.2512.01496,
title = {A Generalization of Caffarelli's Contraction Theorem to Nearly Spherical Manifolds},
author = {Yuxin Ge and Jordan Serres},
journal= {arXiv preprint arXiv:2512.01496},
year = {2025}
}