On the regularity of Ricci flows coming out of metric spaces

Abstract

We consider smooth, not necessarily complete, Ricci flows, $(M,g(t){)}_{t\in (0,T)}$ with $\mathrm{Ric}(g(t))\ge -1$ and $|\mathrm{Rm}(g(t))|\le c/t$ for all $t\in (0,T)$ coming out of metric spaces $(M,d_0)$ in the sense that $(M,d(g(t)),x_0)\to (M,d_0,x_0)$ as $t\searrow 0$ in the pointed Gromov-Hausdorff sense. In the case that $B_{g(t)}(x_0,1)⋐M$ for all $t\in (0,T)$ and $d_0$ is generated by a smooth Riemannian metric in distance coordinates, we show using Ricci-harmonic map heat flow, that there is a corresponding smooth solution $\tilde{g}(t{)}_{t\in (0,T)}$ to the $\delta$-Ricci-DeTurck flow on an Euclidean ball ${\mathbb{B}}_r(p_0)\subset {\mathbb{R}}^n$, which can be extended to a smooth solution defined for $t\in [0,T)$. We further show, that this implies that the original solution $g$ can be extended to a smooth solution on $B_{d_0}(x_0,r/2)$ for $t\in [0,T)$, in view of the method of Hamilton.

37 pages, no figures. Journal version, to appear in JEMS. This version contains a small number of extra clarifications and explanations, partly resulting from comments of the referees