# On the regularity of Ricci flows coming out of metric spaces

# 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 Ric(g(t))\geq−1 and |Rm(g(t))|\leq 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)->(M,d_0,x_0) as t->0 in the pointed Gromov-Hausdorff sense. In the case that B_{g(t)}(x_0,1)\Subset 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 B_r(p_0)\subset 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. This is joint work with Alix Deruelle and Miles Simon.