Paper · Apr 7, 2016

The Ricci flow on a cylinder

Jean Cortissoz, Alexander Murcia

math.DGarXiv:1604.02132
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Abstract

In this paper we study the Ricci flow on surfaces homeomorphic to a cylinder (that is, a product of the circle with a compact interval). We prove longtime existence results, results on the asymptotic behavior of the flow, and we report on an interesting phenomenon: convergence to constant curvature in the normalised flow,under certain assumptions on the initial data, cannot be exponential.

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Submitted
Apr 7, 2016
Subjects
math.DG

Comments: Comments and criticism more than welcome

Ricci flowSurfacesLong-time existence & convergence

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 On the rate of convergence of the p-curve shortening flow (Jean C. Cortissoz, Andrés Galindo and Alexander Murcia, 2017)4Cortissoz 2017 Circle patterns on surfaces of finite topological type (Huabin Ge, Bobo Hua and Ze Zhou, 2019)8Ge 2019 On the one dimensional Logarithmic diffusion equation with nonlinear Robin boundary conditions (Jean Cortissoz and César Reyes, 2021)2Cortissoz 2021 Combinatorial curvature flows for generalized circle packings on surfaces with boundary (Xu Xu and Chao Zheng, 2022)7Xu 2022 All two-dimensional expanding Ricci solitons (Luke T. Peachey and Peter M. Topping, 2023)6Peachey 2023 Singular Ricci Flows on surfaces with boundary and positive scalar curvature (Jean C. Cortissoz and Juan J. Villamarín, 2023)1Cortissoz 2023 Generalized circle patterns on surfaces with cusps (Zhiwen Xiong and Xu Xu, 2025)5Xiong 2025 Ricci flow with metric torsion on surfaces of positive Euler characteristic (Shubham Dwivedi, 2026)3Dwivedi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singular Ricci Flows on surfaces with boundary and positive scalar curvature

    Jean C. Cortissoz and Juan J. Villamarín

    Topics Surfaces · Same author Jean Cortissoz

  2. On the one dimensional Logarithmic diffusion equation with nonlinear Robin boundary conditions

    Jean Cortissoz and César Reyes

    Topics Long-time existence & convergence · Same author Jean Cortissoz · Shared terms “initial data”

  3. Ricci flow with metric torsion on surfaces of positive Euler characteristic

    Shubham Dwivedi

    Topics Surfaces, Long-time existence & convergence

  4. On the rate of convergence of the -curve shortening flow

    Jean C. Cortissoz, Andrés Galindo and Alexander MurciaWider flows

    Same authors Jean Cortissoz, Alexander Murcia

  5. Generalized circle patterns on surfaces with cusps

    Zhiwen Xiong and Xu Xu

    Topics Surfaces, Long-time existence & convergence · Shared terms “longtime existence”

  6. All two-dimensional expanding Ricci solitons

    Luke T. Peachey and Peter M. Topping

    Topics Surfaces · Shared terms “interval”, “initial data”

  7. Combinatorial curvature flows for generalized circle packings on surfaces with boundary

    Xu Xu and Chao Zheng

    Topics Surfaces, Long-time existence & convergence · Shared terms “longtime existence”

  8. Circle patterns on surfaces of finite topological type

    Huabin Ge, Bobo Hua and Ze Zhou

    Topics Surfaces, Long-time existence & convergence

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 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.