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Ricci flow from metrics with isolated conical singularities

Geometric Flows in Riemannian and Complex Geometry May 02, 2016 - May 06, 2016

May 02, 2016 (11:00 AM PDT - 12:00 PM PDT)
Speaker(s): Felix Schulze (University College)
Location: MSRI: Simons Auditorium
Tags/Keywords
• complex geometry

• Riemannian geometry

• geometric analysis

• geometric flow

• Ricci flow

• singularities of flows

• conical singularities

• rescaling of solutions

• positive curvature

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

14495

Abstract

Let $(M,g_0)$ be a compact n-dimensional Riemannian manifold with a finite number of singular points, where at each singular point the metric is asymptotic to a cone over a compact (n-1)-dimensional manifold with curvature operator greater or equal to one. 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. To construct this solution, we desingularize the initial metric by glueing in expanding solitons with positive curvature operator, each asymptotic to the cone at the singular point, at a small scale s. Localizing a recent stability result of Deruelle-Lamm for such expanding solutions, we show that there exists a solution from the desingularized initial metric for a uniform time T>0, independent of the glueing scale s. The solution is then obtained by letting s->0. We also show that the so obtained limiting solution has the corresponding expanding soliton as a forward tangent flow at each initial singular point. This is joint work with P. Gianniotis

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