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

### Alix Deruelle

Institut de Mathématiques de Jussieu, Paris, France### Felix Schulze

University of Warwick, Coventry, UK### Miles Simon

University of Magdeburg, Germany

## Abstract

We consider smooth, possibly incomplete, $n$-dimensional Ricci flows $(M,g(t))_{t\in (0,T)}$ with ${\mathrm{Ric}}(g(t)) \geq -1$ and $| {\mathrm{Rm}} (g(t))| \leq c/t$ for all $t\in (0 ,T)$ {\it 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 $B_{g(t)}(x_0,1) \Subset M$ 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.

## Cite this article

Alix Deruelle, Felix Schulze, Miles Simon, On the regularity of Ricci flows coming out of metric spaces. J. Eur. Math. Soc. 24 (2022), no. 7, pp. 2233–2277

DOI 10.4171/JEMS/1138