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

TL;DR

This paper extends Caffarelli's contraction theorem to nearly spherical manifolds by demonstrating they can be obtained from a round sphere through volume-preserving optimal transport maps, providing new insights into geometric analysis.

Contribution

It introduces a stability result for optimal transport maps on the sphere and generalizes Caffarelli's theorem to nearly spherical manifolds.

Findings

Nearly spherical manifolds can be realized as volume-preserving images of the sphere.

A stability result for optimal transport maps on the sphere is established.

The work provides a perturbative proof of Milman's conjecture.

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.