On the regularity of Ricci flows coming out of metric spaces

Alix Deruelle; Felix Schulze; Miles Simon

Paper · Apr 26, 2019

On the regularity of Ricci flows coming out of metric spaces

Alix Deruelle, Felix Schulze, Miles Simon

math.DGarXiv:1904.11870v2

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Abstract

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 $(M, d (g (t)), x_{0}) \to (M, d_{0}, x_{0})$ as $t ↘ 0$ in the pointed Gromov-Hausdorff sense. In the case that $B_{g (t)} (x_{0}, 1) ⋐ M$ 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 $g ~ (t)_{t \in (0, T)}$ to the $δ$ -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.

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Submitted: Apr 26, 2019
Last revised: Jun 4, 2020 (v2)
Subjects: math.DG, math.AP

Comments: 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

Ricci flow Weak & singular flows Limit spaces & RCD Harmonic maps & DeTurck trick

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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

In the guidebook

6C.11
Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods
Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.
9B.4
Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds
Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.
11B.3
Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities
Hamilton’s compactness theorem and the missing injectivity radius bound.

[1] 6C.11
Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods
Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

[2] 9B.4
Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds
Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

[3] 11B.3
Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities
Hamilton’s compactness theorem and the missing injectivity radius bound.

On the regularity of Ricci flows coming out of metric spaces

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