The Ricci flow on the sphere with marked points
Differential Geometry
2014-07-07 v1 Complex Variables
Abstract
The Ricci flow on the 2-sphere with marked points is shown to converge in all three stable, semi-stable, and unstable cases. In the stable case, the flow was known to converge without any reparametrization, and a new proof of this fact is given. The semi-stable and unstable cases are new, and it is shown that the flow converges in the Gromov-Hausdorff topology to a limiting metric space which is also a 2-sphere, but with different marked points and hence a different complex structure. The limiting metric space carries a unique conical constant curvature metric in the semi-stable case, and a unique conical shrinking gradient Ricci soliton in the unstable case.
Keywords
Cite
@article{arxiv.1407.1118,
title = {The Ricci flow on the sphere with marked points},
author = {D. H. Phong and Jian Song and Jacob Sturm and Xiaowei Wang},
journal= {arXiv preprint arXiv:1407.1118},
year = {2014}
}