JournalsjemsVol. 24, No. 7pp. 2233–2277

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
On the regularity of Ricci flows coming out of metric spaces cover
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Abstract

We consider smooth, possibly incomplete, nn-dimensional Ricci flows (M,g(t))t(0,T)(M,g(t))_{t\in (0,T)} with Ric(g(t))1{\mathrm{Ric}}(g(t)) \geq -1 and Rm(g(t))c/t| {\mathrm{Rm}} (g(t))| \leq c/t for all t(0,T)t\in (0 ,T) {\it coming out} of metric spaces (M,d0)(M,d_0) in the sense that (M,d(g(t)),x0)(M,d0,x0)(M,d(g(t)), x_0) \to (M,d_0, x_0) as t0t\searrow 0 in the pointed Gromov–Hausdorff sense. If Bg(t)(x0,1)MB_{g(t)}(x_0,1) \Subset M for all t(0,T)t\in (0,T) and (Bd0(x0,1),d0)(B_{d_0}(x_0,1),d_0) can be isometrically and compactly embedded in a smooth nn-dimensional Riemannian manifold (Ω,d(g~0))(\Omega,d(\tilde{g}_0)), then we show using the Ricci-harmonic map heat flow that there is a corresponding smooth solution (g~(t))t(0,T)(\tilde g(t))_{t\in (0,T)} to the δ\delta-Ricci–DeTurck flow on a Euclidean ball Br(p0)Rn{\mathbb B}_{r}(p_0) \subset {\mathbb R}^n, for some small 0<r<10<r<1, and g~(t)g~0\tilde{g}(t) \to \tilde{g}_0 smoothly as t0t\to 0. We further show that this implies that the original solution gg 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