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Ricci flow from spaces with isolated conical singularities

Panagiotis Gianniotis and Felix Schulze

Geometry & Topology 22 (2018) 3925–3977

Let (M,g0) 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 (Sn1,g). We show that there exists a smooth Ricci flow starting from such a metric with curvature decaying like Ct. 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 (Sn1Γ,g), where Γ 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.

Ricci flow, singular initial data, conical singularities
Mathematical Subject Classification 2010
Primary: 53C44
Secondary: 58J47
Received: 9 January 2017
Revised: 21 February 2018
Accepted: 1 July 2018
Published: 6 December 2018
Proposed: Bruce Kleiner
Seconded: Gang Tian, John Lott
Panagiotis Gianniotis
Department of Mathematics
University of Toronto
Toronto, ON
Felix Schulze
Department of Mathematics
University College London
United Kingdom