A Flow Tangent to the Ricci Flow via Heat Kernels and Mass Transport

TL;DR

This paper establishes a novel connection between heat flow, optimal transport, and Ricci flow, enabling the evolution of metrics on non-smooth spaces with Ricci curvature bounds.

Contribution

It introduces a new relation linking heat kernels, optimal transport, and Ricci flow, and extends Ricci flow concepts to non-smooth metric measure spaces.

Findings

New relation between heat flow and Ricci flow

Framework for evolving metrics on non-smooth spaces

Potential applications in geometric analysis

Abstract

We present a new relation between the short time behavior of the heat flow, the geometry of optimal transport and the Ricci flow. We also show how this relation can be used to define an evolution of metrics on non-smooth metric measure spaces with Ricci curvature bounded from below.