Paper · Aug 22, 2018

Notes on Ricci flows with collapsing initial data (I): Distance distortion

Shaosai Huang

math.DGarXiv:1808.07394v2
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Abstract

In this note, we prove a uniform distance distortion estimate for Ricci flows with uniformly bounded scalar curvature, independent of the lower bound of the initial -entropy. Our basic principle tells that once correctly renormalized, the metric-measure quantities obey similar estimates as in the non-collapsing case; espeically, the lower bounds of the renormalized heat kernel, observed on a scale comparable to the initial diameter, matches with the lower bound of the renormalized volume ratio, giving the desired distance distortion estimate.

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Submitted
Aug 22, 2018
Last revised
Sep 6, 2018 (v2)
Subjects
math.DG

Comments: A corollary added, a reference added, discussion shortened

Ricci flow(Non)collapsingHeat kernelScalar curvatureLimit spaces & RCD

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 Heat kernel on Ricci shrinkers (Yu Li and Bing Wang, 2019)6Li 2019 Rigidity of the first Betti number via Ricci flow smoothing (Shaosai Huang and Bing Wang, 2020)2Huang 2020 Ricci flow smoothing for locally collapsing manifolds (Shaosai Huang and Bing Wang, 2020)1Huang 2020 Ricci limit flows and weak solutions (Beomjun Choi and Robert Haslhofer, 2021)7Choi 2021 Heat kernel on Ricci shrinkers (II) (Yu Li and Bing Wang, 2023)4Li 2023 Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow (Shaochuang Huang and Zhuo Peng, 2026)8Huang 2026 The Singular Fibration of Collapsing RCD Spaces with Reifenberg Local Covers (Shaosai Huang, 2026)5Huang 2026 Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits (Lei Zhang and Zhenlei Zhang, 2026)3Zhang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci flow smoothing for locally collapsing manifolds

    Shaosai Huang and Bing Wang

    Topics (Non)collapsing · Same author Shaosai Huang · Shared terms “collapsing initial”, “distance distortion”, “distortion estimate”

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

    Shaosai Huang and Bing Wang

    Topics (Non)collapsing, Limit spaces & RCD · Same author Shaosai Huang · Shared terms “collapsing initial”, “initial data”, “uniformly bounded”

  3. Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits

    Lei Zhang and Zhenlei Zhang

    Topics Heat kernel, (Non)collapsing, Scalar curvature · Shared terms “distance distortion”

  4. Heat kernel on Ricci shrinkers (II)

    Yu Li and Bing Wang

    Topics Heat kernel, Limit spaces & RCD

  5. The Singular Fibration of Collapsing RCD Spaces with Reifenberg Local Covers

    Shaosai Huang

    Topics (Non)collapsing, Limit spaces & RCD · Same author Shaosai Huang

  6. Heat kernel on Ricci shrinkers

    Yu Li and Bing Wang

    Topics Heat kernel, (Non)collapsing

  7. Ricci limit flows and weak solutions

    Beomjun Choi and Robert Haslhofer

    Topics Heat kernel, (Non)collapsing, Limit spaces & RCD

  8. Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow

    Shaochuang Huang and Zhuo Peng

    Topics (Non)collapsing, Limit spaces & RCD, Scalar curvature · Shared terms “lower bound”

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) Heat kernel The heat kernel of the conjugate heat equation, coupled to Ricci flow, is the probe Perelman and Bamler use to measure geometry. Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986) 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)

In the guidebook

  1. 6A.3
    The Heat Equation on ℝⁿBook 6A · The Heat Equation and Its Relatives

    The heat kernel, smoothing, uniqueness, Brownian motion, and a Gaussian that runs to Perelman.

  2. 6A.4
    Fourier Methods and Scale SpaceBook 6A · The Heat Equation and Its Relatives

    Fourier's own method on the circle and the line, smoothing as the decay of high frequencies, and the heat equation as the mathematics of blurring images.

  3. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.