Paper · Jun 26, 2017

The local entropy along Ricci flow—Part A: the no-local-collapsing theorems

Bing Wang

math.DGarXiv:1706.08485
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As on arXiv

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 estimate along the Ricci flow. It can be used to show the Gromov-Hausdorff convergence of the Kähler Ricci flow on each minimal projective manifold of general type.

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Submitted
Jun 26, 2017
Subjects
math.DG, math.AG

Comments: 63 pages, 10 figures, comments welcome!

Ricci flowKähler–Ricci flow(Non)collapsingHarnack inequalitiesKähler geometryLimit spaces & RCDAnalytic minimal model program

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 Remarks of weak-compactness along Kahler Ricci flow (Xiuxiong Chen and Bing Wang, 2016)5Chen 2016 The Rigidity of Ricci shrinkers of dimension four (Yu Li and Bing Wang, 2017)8Li 2017 Relative volume comparison of Ricci Flow and its applications (Gang Tian and Zhenlei Zhang, 2018)4Tian 2018 On the structure of Ricci shrinkers (Haozhao Li, Yu Li and Bing Wang, 2018)7Li 2018 Heat kernel on Ricci shrinkers (Yu Li and Bing Wang, 2019)3Li 2019 Rigidity of the first Betti number via Ricci flow smoothing (Shaosai Huang and Bing Wang, 2020)6Huang 2020 The local entropy along Ricci flow—Part B: the pseudo-locality theorems (Bing Wang, 2020)1Wang 2020 On the improved no-local-collapsing theorem of Ricci flow (Wangjian Jian, 2021)2Jian 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 B: the pseudo-locality theorems

    Bing Wang

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

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

    Wangjian Jian

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

  3. Heat kernel on Ricci shrinkers

    Yu Li and Bing Wang

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

  4. Relative volume comparison of Ricci Flow and its applications

    Gang Tian and Zhenlei Zhang

    Topics Kähler–Ricci flow, (Non)collapsing, Limit spaces & RCD · Shared terms “local collapsing”, “along Ricci”, “estimate along”

  5. Remarks of weak-compactness along Kahler Ricci flow

    Xiuxiong Chen and Bing Wang

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same author Bing Wang

  6. 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 “Gromov Hausdorff”

  7. On the structure of Ricci shrinkers

    Haozhao Li, Yu Li and Bing Wang

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

  8. The Rigidity of Ricci shrinkers of dimension four

    Yu Li and Bing Wang

    Topics Harnack inequalities, Limit spaces & RCD · Same author Bing Wang · Shared terms “Gromov Hausdorff”

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) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) (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)

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.