Paper · Aug 20, 2024

Removing scalar curvature assumption for Ricci flow smoothing

Adam Martens

math.DGarXiv:2408.11115v2
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Abstract

In recent work of Chan-Huang-Lee, it is shown that if a manifold enjoys uniform bounds on (a) the negative part of the scalar curvature, (b) the local entropy, and (c) volume ratios up to a fixed scale, then there exists a Ricci flow for some definite time with estimates on the solution assuming that the local curvature concentration is small enough initially (depending only on these a priori bounds). In this work, we show that the bound on scalar curvature assumption (a) is redundant. We also give some applications of this quantitative short-time existence, including a Ricci flow smoothing result for measure space limits, a Gromov-Hausdorff compactness result, and a topological and geometric rigidity result in the case that the a priori local bounds are strengthened to be global.

Comments: Published in Bulletin of the London Mathematical Society. Final version

Ricci flowSingularitiesScalar curvatureLimit spaces & RCDStability & uniquenessHarmonic maps & DeTurck trick

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    Topics Harmonic maps & DeTurck trick, Limit spaces & RCD, Stability & uniqueness · Shared terms “smoothing”, “Gromov Hausdorff”, “short time”

  2. A note on a diffeomorphism criterion via long-time Ricci flow

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    Topics Scalar curvature · Shared terms “Removing scalar”, “flow smoothing”, “curvature concentration”

  3. Sharpening a gap theorem: nonnegative Ricci and small curvature concentration

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    Topics Ricci flow · Same author Adam Martens · Shared terms “curvature concentration”, “volume ratio”, “small enough”

  4. Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition

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    Topics Harmonic maps & DeTurck trick, Scalar curvature, Singularities · Shared terms “flow smoothing”, “short time”, “time existence”

  5. Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds

    Shouhei Honda, Christian Ketterer, Ilaria Mondello et al. PDF

    Topics Harmonic maps & DeTurck trick, Limit spaces & RCD, Scalar curvature · Shared terms “Gromov Hausdorff”

  6. Ricci-harmonic flow of and Spin(7)-structures

    Shubham Dwivedi PDF

    Topics Harmonic maps & DeTurck trick, Limit spaces & RCD, Stability & uniqueness · Shared terms “uniform bounds”, “short time”, “compactness”

  7. Ricci flow on Finsler manifolds

    Behroz Bidabad and Maral K. Sedaghat PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

  8. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam PDF

    Topics Harmonic maps & DeTurck trick, Scalar curvature, Stability & uniqueness · Shared terms “Gromov”

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Background reading

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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)

In the guidebook

  1. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  2. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

  3. 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.