An energy approach to uniqueness for higher-order geometric flows

TL;DR

This paper introduces an energy-based method to prove the uniqueness of solutions for a wide range of higher-order geometric flows, avoiding traditional techniques like the DeTurck trick.

Contribution

It extends an energy argument approach to all orders of curvature flows, including quasilinear and fully nonlinear flows, broadening the scope of uniqueness proofs in geometric analysis.

Findings

Proves uniqueness for higher-order curvature flows using energy methods.

Applies the approach to the fully nonlinear cross-curvature flow.

Extends previous techniques to a broader class of geometric evolution equations.

Abstract

We demonstrate that the uniqueness of solutions to a broad class of parabolic geometric evolution equations can be proven via a direct and essentially classical energy argument which avoids the DeTurck trick entirely. Previously, we have used a variation of this technique to give an alternative proof and slight extension to the basic uniqueness result for complete solutions to the Ricci flow of uniformly bounded curvature. Here we extend this approach to curvature flows of all orders, including the $L^2$-curvature flow and a class of quasilinear higher-order flows related to the obstruction tensor. We also detail its application to the fully nonlinear cross-curvature flow.