Paper · Mar 13, 2016

Convergence of Ricci flows with bounded scalar curvature

Richard H. Bamler

math.DGarXiv:1603.05235v2
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Abstract

In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of codimension . We also establish a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case. These results are based on a compactness theorem for Ricci flows with bounded scalar curvature, which states that any sequence of such Ricci flows converges, after passing to a subsequence, to a metric space that is smooth away from a set of codimension . In the course of the proof, we will also establish -curvature bounds on time-slices of such flows.

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Submitted
Mar 13, 2016
Last revised
Feb 7, 2018 (v2)
Subjects
math.DG, math.AP

Comments: 71 pages, formerly part of arXiv:1512.08527

Ricci flowScalar curvatureLimit spaces & RCDLong-time existence & convergence

Its neighbourhood

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

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  1. Structure theory of non-collapsed limits of Ricci flows

    Richard H Bamler

    Topics Limit spaces & RCD, Long-time existence & convergence · Same author Richard H. Bamler · Shared terms “smooth away”, “singular set”, “curvature bounds”

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  3. Compactness theory of the space of Super Ricci flows

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    Topics Limit spaces & RCD · Same author Richard H. Bamler · Shared terms “compactness”, “curvature bounds”, “sequence”

  4. Structure theory of singular spaces

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    Topics Limit spaces & RCD · Same author Richard H. Bamler · Shared terms “sequence”

  5. Type of finite time singularities of the Ricci flow with bounded scalar curvature

    Shota Hamanaka

    Topics Scalar curvature · Shared terms “bounded scalar”, “sequence”

  6. Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7

    Xianbin Huang and Wancheng Zhang

    Topics Scalar curvature · Shared terms “bounded scalar”, “curvature bounds”

  7. Entropy and heat kernel bounds on a Ricci flow background

    Richard H Bamler

    Topics Limit spaces & RCD, Scalar curvature · Same author Richard H. Bamler · Shared terms “compactness”, “curvature bounds”

  8. Smoothing polyhedral spaces via Ricci flow

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    Topics Limit spaces & RCD · Same author Richard H. Bamler · Shared terms “time slice”, “curvature bounds”

Read around it

Background reading

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

Foundational paper in the Kähler–Ricci flow guide: Bamler: convergence of Ricci flows with bounded scalar curvature; an independent proof of Hamilton–Tian.

Foundational paper in the Kähler–Ricci solitons guide: Bamler: an independent proof of the Hamilton–Tian conjecture.

Foundational paper in the Long-time existence & convergence guide: Bamler: convergence of flows with bounded scalar curvature.

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  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.