A mechanics for the Ricci flow

TL;DR

This paper links the Ricci flow on conformally flat manifolds to a classical mechanics framework by deriving a Hamilton-Jacobi equation that describes the flow.

Contribution

It introduces a novel mechanical interpretation of the Ricci flow via a Hamilton-Jacobi formulation for conformally flat metrics.

Findings

Ricci flow corresponds to a time-dependent Hamilton-Jacobi equation.

Provides a mechanical perspective on geometric evolution equations.

Establishes a new connection between differential geometry and classical mechanics.

Abstract

We construct the classical mechanics associated with a conformally flat Riemannian metric on a compact, n-dimensional manifold without boundary. The corresponding gradient Ricci flow equation turns out to equal the time-dependent Hamilton-Jacobi equation of the mechanics so defined.