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

Yuxin Ge, Jordan Serres

TL;DR

The paper extends Caffarelli's contraction theorem to nearly spherical manifolds by realizing such spaces as volume-preserving images of a round sphere via the Brenier–McCann optimal transport map. It proves a perturbative Milman conjecture in arbitrary dimensions, showing that for small $C^{0,\alpha}$ perturbations of the sphere, the optimal transport map from the round sphere to the perturbed sphere is 1-Lipschitz. Non-quantitative stability results for sphere-based OT maps are established, providing robustness of contraction under perturbations and highlighting the role of MTW regularity and away-from-cut-locus estimates. These results contribute to extending contraction-type transport phenomena to CD(ρ,n) spaces and deepen the understanding of OT stability on curved manifolds, with implications for diameter, isoperimetric, and spectral inequalities in nearly spherical geometries.

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.

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

TL;DR

The paper extends Caffarelli's contraction theorem to nearly spherical manifolds by realizing such spaces as volume-preserving images of a round sphere via the Brenier–McCann optimal transport map. It proves a perturbative Milman conjecture in arbitrary dimensions, showing that for small perturbations of the sphere, the optimal transport map from the round sphere to the perturbed sphere is 1-Lipschitz. Non-quantitative stability results for sphere-based OT maps are established, providing robustness of contraction under perturbations and highlighting the role of MTW regularity and away-from-cut-locus estimates. These results contribute to extending contraction-type transport phenomena to CD(ρ,n) spaces and deepen the understanding of OT stability on curved manifolds, with implications for diameter, isoperimetric, and spectral inequalities in nearly spherical geometries.

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.

Paper Structure

This paper contains 4 sections, 7 theorems, 45 equations.

Table of Contents

  1. Introduction and main results
  2. Related literature
  3. Proof of Theorem \ref{['thm:main']}
  4. Proof of the stability Theorems \ref{['thm:stability']} and \ref{['thm:stabilitybis']}

Key Result

Theorem 1.2

Let $\alpha\in(0,1)$, let $\mathbb{S}^n$ be the standard differentiable $n$-sphere, let $g_\rho$ be the metric of the round sphere of radius $\rho\in(0,1)$, and let $g_{can}$ be the canonical metric of the round sphere of radius $1$. There exists $\varepsilon>0$ only depending on the dimension $n\ge the optimal transport map $T: \mathbb{S}^n\to\mathbb{S}^n$ for the quadratic cost $d_{g_{can}}^2$,

Theorems & Definitions (11)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 1 more