On the regularity of Ricci flows coming out of metric spaces

We consider smooth, not necessarily complete, Ricci flows, $(M,g(t)){t\in (0,T)}$with${$\mathrm{R}\mathrm{i}\mathrm{c}$}(g(t)) \geq -1$and$| {$\mathrm{R}\mathrm{m}$} (g(t))| \leq c/t$forall$t\in (0 ,T)$comingoutofmetricspaces$(M,d_0)$ 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 ${\mathbb B}{r}(p_0) \subset {\mathbb R}^n$,whichcanbeextendedtoasmoothsolutiondefinedfor$t \in [0,T)$\mathrm{.}Wefurthershow,thatthisimpliesthattheoriginalsolution$g$ can be extended to a smooth solution on $B_{d_0}(x_0,r\mathrm{/}2)$ for $t\in [0,T)$, in view of the method of Hamilton.