Paper · May 28, 2016

Analysis aspects of Ricci flow on conical surfaces

Hao Yin

math.AParXiv:1605.08836
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Abstract

In this paper, we establish a framework for the analysis of linear parabolic equations on conical surfaces and use them to study the conical Ricci flow. In particular, we prove the long time existence of the conical Ricci flow for general cone angle and show that this solution has the optimal regularity, namely, the time derivatives of the conformal factor are bounded and for each fixed time, the conformal factor has an explicit asymptotic expansion near the cone points.

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Submitted
May 28, 2016
Subjects
math.AP, math.DG
Ricci flowLong-time existence & convergence

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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 conical Kähler-Ricci flow with weak initial data on Fano manifold (Jiawei Liu and Xi Zhang, 2016)6Liu 2016 On the deformation of discrete conformal factors on surfaces (Huabin Ge and Wenshuai Jiang, 2016)5Ge 2016 Rate of curvature decay for the contracting cusp Ricci flow (Peter M. Topping and Hao Yin, 2016)3Topping 2016 Smooth approximation of the modified conical Kähler-Ricci flow (Ryosuke Takahashi, 2017)4Takahashi 2017 Smoothing a measure on a Riemann surface using Ricci flow (Peter M. Topping and Hao Yin, 2021)2Topping 2021 Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces (Peter M. Topping and Hao Yin, 2023)1Topping 2023 Long-time existence of some geometric flows with bounded scalar curvature (Chuanhuan Li and Yi Li, 2026)7Li 2026 On the geometry of asymptotically conical Ricci shrinkers (I): weighted analysis (Yu Li, Longhuan Yan and Wenjia Zhang, 2026)8Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces

    Peter M. Topping and Hao Yin

    Topics Ricci flow · Same author Hao Yin · Shared terms “conformal factor”

  2. Smoothing a measure on a Riemann surface using Ricci flow

    Peter M. Topping and Hao Yin

    Topics Ricci flow · Same author Hao Yin

  3. Rate of curvature decay for the contracting cusp Ricci flow

    Peter M. Topping and Hao Yin

    Topics Ricci flow · Same author Hao Yin

  4. Smooth approximation of the modified conical Kähler-Ricci flow

    Ryosuke Takahashi

    Topics Long-time existence & convergence

  5. On the deformation of discrete conformal factors on surfaces

    Huabin Ge and Wenshuai JiangWider flows

    Topics Long-time existence & convergence · Shared terms “conformal factor”

  6. The conical Kähler-Ricci flow with weak initial data on Fano manifold

    Jiawei Liu and Xi Zhang

    Topics Long-time existence & convergence

  7. Long-time existence of some geometric flows with bounded scalar curvature

    Chuanhuan Li and Yi Li

    Topics Long-time existence & convergence

  8. On the geometry of asymptotically conical Ricci shrinkers (I): weighted analysis

    Yu Li, Longhuan Yan and Wenjia Zhang

    Shared terms “conical Ricci”

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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… 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.