Ricci flow from spaces with isolated conical singularities

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 $\left({\mathbb{S}}^{n-1}∕\mathrm{\Gamma },g\right)$, where $\mathrm{\Gamma }$ 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.