Vol. 8, No. 4, 2015

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Ricci flow on surfaces with conic singularities

Rafe Mazzeo, Yanir A. Rubinstein and Natasa Sesum

Vol. 8 (2015), No. 4, 839–882
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 $2\pi$, 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.

Keywords
Ricci flow, conic singularities, heat kernels
Mathematical Subject Classification 2010
Primary: 53C44, 58J35
Milestones
Received: 26 May 2014
Revised: 27 January 2015
Accepted: 6 March 2015
Published: 21 June 2015
Authors
 Rafe Mazzeo Department of Mathematics Stanford University Stanford, CA 94305 United States Yanir A. Rubinstein Department of Mathematics University of Maryland College Park, MD 20742 United States Natasa Sesum Department of Mathematics Rutgers University 110 Frelinghuysen road Piscataway, NJ 10027 United States