A short proof of backward uniqueness for some geometric evolution equations

TL;DR

This paper presents a straightforward proof of backward uniqueness for certain geometric evolution equations, including Ricci and cross-curvature flows, using a classical energy convexity approach instead of Carleman inequalities.

Contribution

It introduces a simple, direct proof technique based on logarithmic convexity, applicable to various second-order and higher-order geometric flows.

Findings

Proves backward uniqueness for Ricci and cross-curvature flows.

Extends the technique to the $L^2$-curvature flow and other higher-order equations.

Demonstrates the method's broad applicability to geometric evolution equations.

Abstract

We give a simple, direct proof of the backward uniqueness of solutions to a class of second-order geometric evolution equations including the Ricci and cross-curvature flows. The proof, based on a classical argument of Agmon-Nirenberg, uses the logarithmic convexity of a certain energy quantity in the place of Carleman inequalities. We also demonstrate the applicability of the technique to the $L^2$-curvature flow and other higher-order equations.