Uniqueness of Weak Solutions for the Normalised Ricci Flow in Two Dimensions

TL;DR

This paper proves the uniqueness of classical solutions to the normalized 2D Ricci flow on closed manifolds under minimal regularity assumptions, contrasting with known non-uniqueness in harmonic map heat flow.

Contribution

It establishes the uniqueness of solutions with only a uniform Liouville energy bound and an $L^2$-bound on the time derivative, a novel result in geometric analysis.

Findings

Uniqueness holds under minimal regularity assumptions.

Contrasts with non-uniqueness in harmonic map heat flow.

Provides new insights into 2D Ricci flow behavior.

Abstract

We show uniqueness of classical solutions of the normalised two-dimensional Hamilton-Ricci flow on closed, smooth manifolds for smooth data among solutions satisfying (essentially) only a uniform bound for the Liouville energy and a natural space-time $L^2$-bound for the time derivative of the solution. The result is surprising when compared with results for the harmonic map heat flow, where non-uniqueness through reverse bubbling may occur.