Ricci flow on surfaces with conic singularities
Abstract
We establish the short-time existence of the Ricci flow on surfaces with a finite number of conic points, all with cone angle between 0 and , where the cone angles remain fixed or change in some smooth prescribed way. For the angle-preserving flow we prove long-time existence and convergence. When the Troyanov angle condition is satisfied (equivalently, when the data is logarithmically K-stable), the flow converges to the unique constant curvature metric with the given cone angles; if this condition is not satisfied, the flow converges subsequentially to a soliton. This is the one-dimensional version of the Hamilton--Tian conjecture.
Keywords
Cite
@article{arxiv.1306.6688,
title = {Ricci flow on surfaces with conic singularities},
author = {Rafe Mazzeo and Yanir A. Rubinstein and Natasa Sesum},
journal= {arXiv preprint arXiv:1306.6688},
year = {2015}
}
Comments
v1: 38 pages v2: 39 pages, restructured Sections 1 and 2, and added references and Subsection 5.4. v3-v4: 41 pages, revised to address referee comments; original proof of Proposition 5.3 had an error pointed out to us by a referee. We fix this by invoking Chow and Hamilton's original arguments instead of the Hamilton compactness theorem. Final version. To appear in Analysis and PDE