Ricci flow from spaces with isolated conical singularities

TL;DR

This paper proves the existence of smooth Ricci flows starting from compact manifolds with isolated conical singularities, extending the flow to orbifold singularities, with curvature decaying like C/t and convergence in Gromov-Hausdorff sense.

Contribution

It establishes the existence and construction of Ricci flows from spaces with isolated conical and orbifold singularities, including convergence and regularity results.

Findings

Existence of Ricci flow from manifolds with conical singularities.

Extension of Ricci flow to orbifold singularities.

Curvature decay rate of C/t during the flow.

Abstract

Let $(M,g_0)$ be a compact $n$-dimensional Riemannian manifold with a finite number of singular points, where the metric is asymptotic to a non-negatively curved cone over $(\mathbb{S}^{n-1},g)$. 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 non-negatively curved cone over $(\mathbb{S}^{n-1}/\Gamma,g)$, 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.