#### Volume 22, issue 7 (2018)

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Ricci flow from spaces with isolated conical singularities

### Panagiotis Gianniotis and Felix Schulze

Geometry & Topology 22 (2018) 3925–3977
##### Abstract

Let $\left(M,{g}_{0}\right)$ be a compact $n$–dimensional Riemannian manifold with a finite number of singular points, where the metric is asymptotic to a nonnegatively curved cone over $\left({\mathbb{S}}^{n-1},g\right)$. We show that there exists a smooth Ricci flow starting from such a metric with curvature decaying like $C∕t$. The initial metric is attained in Gromov–Hausdorff distance and smoothly away from the singular points. In the case that the initial manifold has isolated singularities asymptotic to a nonnegatively curved cone over $\left({\mathbb{S}}^{n-1}∕\Gamma ,g\right)$, where $\Gamma$ acts freely and properly discontinuously, we extend the above result by showing that starting from such an initial condition there exists a smooth Ricci flow with isolated orbifold singularities.

##### Keywords
Ricci flow, singular initial data, conical singularities
Primary: 53C44
Secondary: 58J47
##### Publication
Revised: 21 February 2018
Accepted: 1 July 2018
Published: 6 December 2018
Proposed: Bruce Kleiner
Seconded: Gang Tian, John Lott
##### Authors
 Panagiotis Gianniotis Department of Mathematics University of Toronto Toronto, ON Canada Felix Schulze Department of Mathematics University College London London United Kingdom