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Annals of Global Analysis and Geometry

, Volume 29, Issue 4, pp 333–342 | Cite as

Higher order curvature flows on surfaces

  • Hartmut R. Schwetlick
Article
  • 75 Downloads

Abstract

We consider a sixth and an eighth order conformal flow on Riemannian surfaces, which arise as gradient flows for the Calabi energy with respect to a higher order metric. Motivated by a work of Struwe which unified the approach to the Hamilton-Ricci and Calabi flow, we extend the method to these higher order cases. Our results contain global existence and exponentially fast convergence to a constant scalar curvature metric.

Uniform bounds on the conformal factor are obtained via the concentration-compactness result for conformal metrics. In the case of the sphere we use the idea of DeTurck's gauge flow to derive bounds up to conformal transformation.

We prove exponential convergence by showing that the Calabi energy decreases exponentially fast. The problem of the non-trivial kernel in the evolution of the Calabi energy on the sphere is resolved by using Kazdan-Warner's identity.

Keywords

Geometric flows Higher order parabolic equations MSC 53C44 35K25 

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Mathematical SciencesUniversity of BathBathUK

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