Paper · Oct 27, 2022

Pseudolocality theorems of Ricci flows on incomplete manifolds

Liang Cheng

math.DGarXiv:2210.15397v5
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Abstract

In this paper we study the pseudolocality theorems of Ricci flows on incomplete manifolds. We prove that if a ball with its closure contained in an incomplete manifold has the small scalar curvature lower bound and almost Euclidean isoperimetric constant, or almost Euclidean local constant, then we can construct a solution of Ricci flow in the ball which have the pseudolocality property. We also give two applications. First, we prove the short-time existence of Ricci flows on complete manifolds with scalar curvature bounded below uniformly and almost Euclidean isoperimetric inequality holds locally. Second, we show that any complete manifold with nonnegative scalar curvature and Euclidean isoperimetric inequality must be isometric to the Euclidean space.

Comments: 15 pages

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

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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 2021, plus the canon.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On local rigidity theorems with respect to the scalar curvature

    Liang Cheng PDF

    Topics Geometric inequalities, Scalar curvature · Same author Liang Cheng · Shared terms “isoperimetric constant”, “Euclidean”

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

    Liang Cheng and Yongjia Zhang PDF

    Topics Boundaries & non-compact · Same author Liang Cheng · Shared terms “almost Euclidean”, “pseudolocality”, “Euclidean space”

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

    Shaochuang Huang and Luen-Fai Tam PDF

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

  4. On the structure of complete -solitons

    Haozhao Li, Yuanqing Ma and Kai Zheng PDF

    Topics Geometric inequalities, Scalar curvature · Shared terms “pseudolocality theorem”, “almost Euclidean”, “Euclidean local”

  5. The Ricci-DeTurck flow on complete manifolds

    Jing-Bin Cai and Bing Wang PDF

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

  6. ADM mass for metrics and distortion under Ricci-DeTurck flow

    Paula Burkhardt-Guim PDF

    Topics Harmonic maps & DeTurck trick, Scalar curvature, Boundaries & non-compact · Shared terms “nonnegative scalar”

  7. Removing scalar curvature assumption for Ricci flow smoothing

    Adam Martens PDF

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

  8. Ricci flow on Finsler manifolds

    Behroz Bidabad and Maral K. Sedaghat PDF

    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.