An Energy Approach to Uniqueness for Higher-Order Geometric Flows

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Abstract

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.

Original languageEnglish (US)
Pages (from-to)1-25
Number of pages25
JournalJournal of Geometric Analysis
DOIs
StateAccepted/In press - Dec 14 2015

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Keywords

  • Energy methods
  • Geometric evolution equations
  • Uniqueness

ASJC Scopus subject areas

  • Geometry and Topology

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