Paper · Aug 30, 2021

Ricci Flow on Torus Bundles

Dmytro Yeroshkin

math.DGarXiv:2108.13281
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Abstract

In this paper we compute the Ricci flow formulas for invariant metrics on prinicpal -bundles compatible with the connection. Our primary focus is on torus bundles which we use to study a notion of Bakry-Émery Ricci flow as well as Ricci flow on circle bundles over Kähler-Einstein manifolds. The latter application gives us solutions to Ricci flows on Heisenberg groups and implicit solutions to Ricci flows on Berger 3-spheres and several other 3-dimensional manifolds.

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Submitted
Aug 30, 2021
Subjects
math.DG
Ricci flowEinstein metrics3-manifoldsKähler geometryLimit spaces & RCDKähler–Einstein & K-stability

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 2021, 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 Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type (Yasuhiro Nakagawa and Satoshi Nakamura, 2022)8Nakagawa 2022 Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow (Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti, 2023)2Krishnan 2023 A version of Bakry-Émery Ricci flow on a finite graph (Bobo Hua, Yong Lin and Tao Wang, 2024)7Hua 2024 Chern-Ricci flow and t-Gauduchon Ricci-flat condition (Eder M. Correa, Giovane Galindo and Lino Grama, 2025)5Correa 2025 Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds (Frederick Tsz-Ho Fong and Hung Tran, 2026)1Fong 2026 Weighted volume comparison and monotonicity for L^p-bound of Bakry-Émery Ricci curvature (Jintao Ye and Xiaohua Zhu, 2026)6Ye 2026 Singularity Models of Finite-Time Kähler-Ricci Flows (Frederick Tsz-Ho Fong and Hung Tran, 2026)4Fong 2026 Finite-Time Singularities of the Kähler–Ricci Flow on a ℂP^m-Bundle over a Product of Kähler–Einstein Manifolds (Yifan Xiao, 2026)3Xiao 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 CP1-bundles over a Product of Kähler-Einstein Manifolds

    Frederick Tsz-Ho Fong and Hung Tran

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry

  2. Toral symmetries of collapsed ancient solutions to the homogeneous Ricci flow

    Anusha M. Krishnan, Francesco Pediconi and Sammy Sbiti

    Topics Einstein metrics, Limit spaces & RCD · Shared terms “torus bundles”

  3. Finite-Time Singularities of the Kähler–Ricci Flow on a -Bundle over a Product of Kähler–Einstein Manifolds

    Yifan Xiao

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry

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

    Frederick Tsz-Ho Fong and Hung Tran

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Shared terms “circle bundle”

  5. Chern-Ricci flow and t-Gauduchon Ricci-flat condition

    Eder M. Correa, Giovane Galindo and Lino Grama

    Topics Limit spaces & RCD, 3-manifolds, Kähler geometry · Shared terms “torus bundles”

  6. Weighted volume comparison and monotonicity for -bound of Bakry-Émery Ricci curvature

    Jintao Ye and Xiaohua Zhu

    Topics Limit spaces & RCD, Kähler geometry · Shared terms “Émery Ricci”, “Bakry Émery”

  7. A version of Bakry-Émery Ricci flow on a finite graph

    Bobo Hua, Yong Lin and Tao Wang

    Topics Limit spaces & RCD · Shared terms “Émery Ricci”, “Bakry Émery”

  8. Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type

    Yasuhiro Nakagawa and Satoshi Nakamura

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.