Paper · Oct 20, 2020

The local entropy along Ricci flow—Part B: the pseudo-locality theorems

Bing Wang

math.DGarXiv:2010.09981
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Abstract

We localize the entropy functionals of G. Perelman and generalize his no-local-collapsing theorem and pseudo-locality theorem. Our generalization is technically inspired by further development of Li-Yau estimates along the Ricci flow. It has various applications, including to show the continuous dependence of the Ricci flow with respect to the initial metric in Gromov-Hausdorff topology with Ricci curvature bounded below, and to show the compactness of the moduli of Kähler manifolds with bounded scalar curvature and a rough locally almost Euclidean condition.

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Submitted
Oct 20, 2020
Subjects
math.DG

Comments: 108 pages, 5 figures, comments welcome

Ricci flow(Non)collapsingHarnack inequalitiesKähler geometryScalar curvatureLimit spaces & RCD

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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 The Rigidity of Ricci shrinkers of dimension four (Yu Li and Bing Wang, 2017)3Li 2017 The local entropy along Ricci flow—Part A: the no-local-collapsing theorems (Bing Wang, 2017)1Wang 2017 On the structure of Ricci shrinkers (Haozhao Li, Yu Li and Bing Wang, 2018)5Li 2018 Heat kernel on Ricci shrinkers (Yu Li and Bing Wang, 2019)2Li 2019 Rigidity of the first Betti number via Ricci flow smoothing (Shaosai Huang and Bing Wang, 2020)4Huang 2020 Ricci flow smoothing for locally collapsing manifolds (Shaosai Huang and Bing Wang, 2020)8Huang 2020 Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing (Shaosai Huang, Xiaochun Rong and Bing Wang, 2020)6Huang 2020 On the improved no-local-collapsing theorem of Ricci flow (Wangjian Jian, 2021)7Jian 2021
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The local entropy along Ricci flow—Part A: the no-local-collapsing theorems

    Bing Wang

    Topics Harnack inequalities, (Non)collapsing, Limit spaces & RCD · Same author Bing Wang · Shared terms “entropy along”, “local entropy”, “along Ricci”

  2. Heat kernel on Ricci shrinkers

    Yu Li and Bing Wang

    Topics (Non)collapsing · Same author Bing Wang · Shared terms “locality theorem”, “pseudo locality”, “collapsing theorem”

  3. The Rigidity of Ricci shrinkers of dimension four

    Yu Li and Bing Wang

    Topics Harnack inequalities, Limit spaces & RCD, Scalar curvature · Same author Bing Wang · Shared terms “Hausdorff topology”, “bounded scalar”, “Gromov Hausdorff”

  4. Rigidity of the first Betti number via Ricci flow smoothing

    Shaosai Huang and Bing Wang

    Topics (Non)collapsing, Limit spaces & RCD · Same author Bing Wang · Shared terms “curvature bounded”, “Gromov Hausdorff”

  5. On the structure of Ricci shrinkers

    Haozhao Li, Yu Li and Bing Wang

    Topics (Non)collapsing, Limit spaces & RCD · Same author Bing Wang

  6. Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing

    Shaosai Huang, Xiaochun Rong and Bing Wang

    Topics (Non)collapsing, Kähler geometry · Same author Bing Wang · Shared terms “curvature bounded”, “Gromov”, “Yau”

  7. On the improved no-local-collapsing theorem of Ricci flow

    Wangjian Jian

    Topics (Non)collapsing, Scalar curvature · Shared terms “collapsing theorem”, “local collapsing”, “Perelman”

  8. Ricci flow smoothing for locally collapsing manifolds

    Shaosai Huang and Bing Wang

    Topics (Non)collapsing · Same author Bing Wang · Shared terms “curvature bounded”

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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) Harnack inequalities Differential Harnack inequalities compare a solution with itself at different points and times. Hamilton's matrix Harnack inequality says curvature… Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 6D.5
    Diffusions on Manifolds and CurvatureBook 6D · Diffusions, Finance and Computation

    Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12A.6
    𝓛-Geodesics and the Variational FormulasBook 12A · Entropy and Noncollapsing

    Perelman's §7.1 derived line by line: the first and second variation of 𝓛-length, 𝓛-Jacobi fields, adapted fields, Hamilton's Harnack expressions, and the differential inequalities for the reduced distance.