Paper · Oct 6, 2016

Existence and applications of Ricci flows via pseudolocality

Fei He

math.DGarXiv:1610.01735v3
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Abstract

We prove the short-time existence of Ricci flows on complete manifolds with scalar curvature bounded below uniformly, Ricci curvature bounded below by a negative quadratic function, and with almost Euclidean isoperimetric inequality holds locally. In particular, this result applies to manifolds with both Ricci curvature and injectivity radius bounded from below. We also study the short-time behaviour of these solutions which may have unbounded curvature at the initial time, and provide some applications. A key tool is Perelman's pseudolocality theorem.

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Submitted
Oct 6, 2016
Last revised
Oct 13, 2024 (v3)
Subjects
math.DG

Comments: Final version. To appear in CAG Volume 32, No 7, 2024

Ricci flowScalar curvatureBoundaries & non-compactHarmonic maps & DeTurck trickGeometric inequalities

Its neighbourhood

Connected papers

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.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Short-time existence of the Ricci flow on complete, non-collapsed 3-manifolds with Ricci curvature bounded from below (Raphael Hochard, 2016)3Hochard 2016 Existence, Lifespan and Transfer Rate of Ricci Flows on Manifolds with Small Ricci Curvature (Fei He, 2016)2He 2016 Almost euclidean Isoperimetric Inequalities in spaces satisfying local Ricci curvature lower bounds (Fabio Cavalletti and Andrea Mondino, 2017)4Cavalletti 2017 Short-Time Existence Theorem for the CR Torsion Flow (Shu-Cheng Chang and Chin-Tung Wu, 2018)8Chang 2018 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)5Huang 2021 Pseudolocality and uniqueness of Ricci flow on almost Euclidean noncompact manifolds (Liang Cheng and Yongjia Zhang, 2021)7Cheng 2021 Pseudolocality theorems of Ricci flows on incomplete manifolds (Liang Cheng, 2022)1Cheng 2022 The Ricci-DeTurck flow on complete manifolds (Jing-Bin Cai and Bing Wang, 2026)6Cai 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Pseudolocality theorems of Ricci flows on incomplete manifolds

    Liang Cheng

    Topics Harmonic maps & DeTurck trick, Geometric inequalities, Scalar curvature · Shared terms “pseudolocality”, “Euclidean isoperimetric”, “almost Euclidean”

  2. Existence, Lifespan and Transfer Rate of Ricci Flows on Manifolds with Small Ricci Curvature

    Fei He

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact · Same author Fei He · Shared terms “unbounded curvature”, “short time”, “quadratic”

  3. Short-time existence of the Ricci flow on complete, non-collapsed -manifolds with Ricci curvature bounded from below

    Raphael Hochard

    Topics Harmonic maps & DeTurck trick · Shared terms “curvature bounded”, “short time”, “time existence”

  4. Almost euclidean Isoperimetric Inequalities in spaces satisfying local Ricci curvature lower bounds

    Fabio Cavalletti and Andrea Mondino

    Topics Geometric inequalities · Shared terms “Euclidean isoperimetric”, “almost Euclidean”, “isoperimetric inequality”

  5. Short time existence for harmonic map heat flow with time-dependent metrics

    Shaochuang Huang and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact · Shared terms “short time”, “complete manifolds”, “curvature bounded”

  6. The Ricci-DeTurck flow on complete manifolds

    Jing-Bin Cai and Bing Wang

    Topics Harmonic maps & DeTurck trick, Boundaries & non-compact · Shared terms “complete manifolds”, “injectivity radius”, “short time”

  7. Pseudolocality and uniqueness of Ricci flow on almost Euclidean noncompact manifolds

    Liang Cheng and Yongjia Zhang

    Topics Boundaries & non-compact · Shared terms “pseudolocality”, “almost Euclidean”

  8. Short-Time Existence Theorem for the CR Torsion Flow

    Shu-Cheng Chang and Chin-Tung Wu

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

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.

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) 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

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.