On the regularity of Ricci flows coming out of metric spaces

Abstract

We consider smooth, possibly incomplete, $n$-dimensional 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. If $B_{g(t)}(x_0,1)⋐M$ for all $t\in (0,T)$ and $(B_{d_0}(x_0,1),d_0)$ can be isometrically and compactly embedded in a smooth $n$-dimensional Riemannian manifold $(\Omega ,d({\tilde{g}}_0))$, then we show using the 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 a Euclidean ball ${\mathbb{B}}_r(p_0)\subset {\mathbb{R}}^n$, for some small $0<r<1$, and $\tilde{g}(t)\to {\tilde{g}}_0$ smoothly as $t\to 0$. We further show that this implies that the original solution $g$ can be extended locally to a smooth solution defined up to time zero, in view of the method of Hamilton.