An Energy Approach to Uniqueness for Higher-Order Geometric Flows

We describe a simple, direct method to prove the uniqueness of solutions to a broad class of parabolic geometric evolution equations. Our argument, which is based on a prolongation procedure and the consideration of certain natural energy quantities, does not require the solution of any auxiliary parabolic systems. In previous work, we used a variation of this technique to give an alternative proof of the uniqueness of complete solutions to the Ricci flow of uniformly bounded curvature. Here we extend this approach to curvature flows of all orders, including the (Formula presented.)-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.