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

## TL;DR

This paper investigates the regularity and extension of Ricci flows originating from metric spaces, establishing conditions under which these flows can be smoothly extended using Ricci-harmonic map heat flow and Ricci-DeTurck flow techniques.

## Contribution

It introduces a method to extend Ricci flows from metric spaces to smooth solutions via Ricci-harmonic map heat flow and Ricci-DeTurck flow, under specific geometric conditions.

## Key findings

- Smooth extension of Ricci flows from metric spaces is possible under certain curvature bounds.
- The use of Ricci-harmonic map heat flow facilitates the regularity analysis.
- Original Ricci flows can be extended smoothly on smaller balls, leveraging Hamilton's method.

## Abstract

We consider smooth, not necessarily complete, 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 $t\in (0 ,T)$ 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. 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 ${\mathbb B}_{r}(p_0) \subset {\mathbb R}^n$, 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.