## Analysis & PDE

• Anal. PDE
• Volume 8, Number 4 (2015), 839-882.

### Ricci flow on surfaces with conic singularities

#### 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.

#### Article information

Source
Anal. PDE, Volume 8, Number 4 (2015), 839-882.

Dates
Revised: 27 January 2015
Accepted: 6 March 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843115

Digital Object Identifier
doi:10.2140/apde.2015.8.839

Mathematical Reviews number (MathSciNet)
MR3366005

Zentralblatt MATH identifier
1322.53070

#### Citation

Mazzeo, Rafe; Rubinstein, Yanir; Sesum, Natasa. Ricci flow on surfaces with conic singularities. Anal. PDE 8 (2015), no. 4, 839--882. doi:10.2140/apde.2015.8.839. https://projecteuclid.org/euclid.apde/1510843115

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