Ricci flow on surfaces with conic singularities

Rafe Mazzeo, Yanir A. Rubinstein, Natasa Sesum

Research output: Contribution to journalArticle

24 Scopus citations

Abstract

We establish short-time existence of the Ricci flow on surfaces with a finite number of conic points, all with cone angle between 0 and 2π, with cone angles remaining fixed or changing in some smooth prescribed way. For the angle-preserving flow we prove long-time existence; if the angles satisfy the Troyanov condition, this flow converges exponentially to the unique constant-curvature metric with these cone angles; if this condition fails, the conformal factor blows up at precisely one point. These geometric results rely on a new refined regularity theorem for solutions of linear parabolic equations on manifolds with conic singularities. This is proved using methods from geometric microlocal analysis, which is the main novelty of this article.

Original languageEnglish (US)
Pages (from-to)839-882
Number of pages44
JournalAnalysis and PDE
Volume8
Issue number4
DOIs
StatePublished - Jan 1 2015

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All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

Keywords

  • Conic singularities
  • Heat kernels
  • Ricci flow

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