On the regularity of Ricci flows coming out of metric spaces

We consider smooth, not necessarily complete, Ricci flows, (M,g(t))t∈(0,T) with Ric(g(t))≥−1 and |Rm(g(t))|≤c/t for all t∈(0,T) coming out of metric spaces (M,d0) in the sense that (M,d(g(t)),x0)→(M,d0,x0) as t↘0 in the pointed Gromov-Hausdorff sense. In the case that Bg(t)(x0,1)⋐M for all t∈(0,T) and d0 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 g~(t)t∈(0,T) to the δ-Ricci-DeTurck flow on an Euclidean ball Br(p0)⊂Rn, which can be extended to a smooth solution defined for t∈[0,T). We further show, that this implies that the original solution g can be extended to a smooth solution on Bd0(x0,r/2) for t∈[0,T), in view of the method of Hamilton.