On the regularity of Ricci flows coming out of metric spaces
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 GromovHausdorff 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 Ricciharmonic map heat flow, that there is a corresponding smooth solution $\tilde g(t)_{t\in (0,T)}$ to the $\delta$RicciDeTurck 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 DOI:
 10.48550/arXiv.1904.11870
 arXiv:
 arXiv:1904.11870
 Bibcode:
 2019arXiv190411870D
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs;
 53C44;
 53C23;
 58J35;
 35B45;
 35K40;
 35K55
 EPrint:
 37 pages, no figures. Journal version, to appear in JEMS. This version contains a small number of extra clarifications and explanations, partly resulting from comments of the referees