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

Yuxin Ge; Jordan Serres

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arXiv · 2512.01496

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

Yuxin Ge·Jordan Serres

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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.

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Yuxin Ge, Jordan Serres. 2025-12-01. A Generalization of Caffarelli's Contraction Theorem to Nearly Spherical Manifolds. https://arxiv.org/abs/2512.01496

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