Contractive transport maps from $\mathbb{S}^2$ to nearly spherical surfaces with positive Ricci curvature

TL;DR

This paper demonstrates that nearly spherical surfaces with positive Ricci curvature can be obtained from a round sphere via a contractive, volume-preserving map, using Ricci flow, Kim-Milman construction, and Bakry-Émery criterion.

Contribution

It introduces a novel method to construct contractive, volume-preserving maps from spheres to nearly spherical surfaces with positive Ricci curvature.

Findings

Every nearly spherical, positively curved surface is a contractive, volume-preserving image of a round sphere.

The proof integrates Ricci flow, Kim-Milman construction, and Bakry-Émery criterion.

The approach provides a new geometric understanding of positively curved surfaces.

Abstract

We prove that every nearly spherical, positively curved surface is the contractive, volume-preserving image of a round sphere. The proof combines three main tools: the Ricci flow on surfaces, the Kim-Milman construction, and a multiscale Bakry-\'Emery criterion.