Paper · Feb 14, 2018

The Ricci flow for circle bundles over surfaces

Arash Bazdar, Georgios Fotopoulos

math.DGarXiv:1802.05363v2
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Abstract

In this work, we study and solve the normalized Ricci flow equation for circle bundles over surfaces. Moreover, we study the asymptotic behavior of the solutions and their connections to some model geometries.

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Submitted
Feb 14, 2018
Last revised
May 7, 2025 (v2)
Subjects
math.DG
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 Eigenvalues under the Ricci flow of model geometries (Songbo Hou, 2016)3Hou 2016 The Ricci flow on a cylinder (Jean Cortissoz and Alexander Murcia, 2016)5Cortissoz 2016 Ancient Ricci Flow Solutions on Bundles (Peng Lu and Y. K. Wang, 2016)1Lu 2016 Ricci Flow on Torus Bundles (Dmytro Yeroshkin, 2021)2Yeroshkin 2021 Classification of superpotentials for cohomogeneity one Ricci solitons (Qiu Shi Wang, 2024)7Wang 2024 A normalized Ricci flow on surfaces with boundary towards the complete hyperbolic metric (Gang Li, 2025)6Li 2025 The Cauchy problem for gradient generalized Ricci solitons on a bundle gerbe (Severin Bunk, Miguel Pino Carmona and C. S. Shahbazi, 2025)8Bunk 2025 Singularity Models of Finite-Time Kähler-Ricci Flows (Frederick Tsz-Ho Fong and Hung Tran, 2026)4Fong 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient Ricci Flow Solutions on Bundles

    Peng Lu and Y. K. Wang

    Topics Ricci flow · Shared terms “circle bundle”

  2. Ricci Flow on Torus Bundles

    Dmytro Yeroshkin

    Topics Ricci flow · Shared terms “circle bundle”

  3. Eigenvalues under the Ricci flow of model geometries

    Songbo Hou

    Topics Ricci flow · Shared terms “model geometries”, “normalized Ricci”

  4. Singularity Models of Finite-Time Kähler-Ricci Flows

    Frederick Tsz-Ho Fong and Hung Tran

    Topics Ricci flow · Shared terms “circle bundle”

  5. The Ricci flow on a cylinder

    Jean Cortissoz and Alexander Murcia

    Topics Ricci flow · Shared terms “asymptotic behavior”

  6. A normalized Ricci flow on surfaces with boundary towards the complete hyperbolic metric

    Gang Li

    Topics Ricci flow · Shared terms “normalized Ricci”, “asymptotic behavior”

  7. Classification of superpotentials for cohomogeneity one Ricci solitons

    Qiu Shi Wang

    Shared terms “circle bundle”

  8. The Cauchy problem for gradient generalized Ricci solitons on a bundle gerbe

    Severin Bunk, Miguel Pino Carmona and C. S. Shahbazi

    Topics Ricci flow

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.