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Mathematische Annalen

, Volume 353, Issue 2, pp 333–357 | Cite as

Conformal entropy rigidity through Yamabe flows

  • Pablo Suárez-SerratoEmail author
  • Samuel Tapie
Article

Abstract

We introduce two versions of the Yamabe flow which preserve negative scalar-curvature bounds. First we show existence and smooth convergence of solutions to these flows. We then show that a metric with negative scalar curvature is controlled by the Yamabe metrics in the same conformal class with constant extremal scalar curvatures. This implies that the volume entropy of our original metric is controlled by the entropies of these Yamabe metrics. We eventually use these Yamabe flows to prove an entropy-rigidity result: when the Yamabe metric has negative sectional curvature, the entropy of a metric in the same conformal class is extremal if and only if the metric has constant extremal scalar curvature.

Mathematics Subject Classification (2000)

53C21 37A35 53C44 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacional Autonóma de México, Ciudad Universitaria, CoyoacánMexicoMexico
  2. 2.Departament de Matemàtica Aplicada 1Universitat Politècnica de CatalunyaBarcelonaSpain
  3. 3.Laboratoire Jean LerayUniversité de Nantes 2Nantes Cedex 3France

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