Paper · Jul 1, 2021

Area and boundary length of surfaces diffeomorphic to annuli

Tsz-Kiu Aaron Chow

math.DGarXiv:2107.00335
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Abstract

In this paper, we give a proof to a statement in Perelman's paper for finite extinction time of Ricci flow. Our proof draws on different techniques from the one given in Morgan-Tian's exposition and is extrinsic in nature, which relies on the co-area formula instead of the Gauss-Bonnet theorem, and is potentially generalizable to higher dimensions.

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Submitted
Jul 1, 2021
Subjects
math.DG

Comments: 16 pages

Ricci flow

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 Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (Grisha Perelman, 2003)5Perelman 2003 Self-Expanders to Inverse Curvature Flows by Homogeneous Functions (Tsz-Kiu Aaron Chow, Ka-Wing Chow and Frederick Tsz-Ho Fong, 2017)7Chow 2017 Ricci Flow on Manifolds with Boundary with Arbitrary Initial Metric (Tsz-Kiu Aaron Chow, 2020)1Chow 2020 Monotonicity of the Cheeger constant under Ricci flow on spheres (Hollis Williams, 2024)8Williams 2024 Finite extinction time of a family of homogeneous Ricci flows (Roberto Araujo, 2025)4Araujo 2025 A Spinorial Perelman's Functional: Critical Points and Gradient Flow (Tsz-Kiu Aaron Chow and Frederick Tsz-Ho Fong, 2026)6Chow 2026 Small Normal Curvature and Three-Manifold Topology (Tsz-Kiu Aaron Chow and Jingbo Wan, 2026)2Chow 2026 Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow (Tsz-Kiu Aaron Chow, 2026)3Chow 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 on Manifolds with Boundary with Arbitrary Initial Metric

    Tsz-Kiu Aaron Chow

    Topics Ricci flow · Same author Tsz-Kiu Aaron Chow

  2. Small Normal Curvature and Three-Manifold Topology

    Tsz-Kiu Aaron Chow and Jingbo Wan

    Topics Ricci flow · Same author Tsz-Kiu Aaron Chow · Shared terms “diffeomorphic”

  3. Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow

    Tsz-Kiu Aaron Chow

    Topics Ricci flow · Same author Tsz-Kiu Aaron Chow

  4. Finite extinction time of a family of homogeneous Ricci flows

    Roberto Araujo

    Topics Ricci flow · Shared terms “finite extinction”, “extinction time”

  5. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds

    Grisha PerelmanThe canon

    Topics Ricci flow · Shared terms “finite extinction”, “extinction time”

  6. A Spinorial Perelman's Functional: Critical Points and Gradient Flow

    Tsz-Kiu Aaron Chow and Frederick Tsz-Ho Fong

    Same author Tsz-Kiu Aaron Chow · Shared terms “Perelman”

  7. Self-Expanders to Inverse Curvature Flows by Homogeneous Functions

    Tsz-Kiu Aaron Chow, Ka-Wing Chow and Frederick Tsz-Ho FongWider flows

    Same author Tsz-Kiu Aaron Chow

  8. Monotonicity of the Cheeger constant under Ricci flow on spheres

    Hollis Williams

    Topics Ricci flow · Shared terms “surface diffeomorphic”

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)

In the guidebook

  1. 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.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.