Paper · Apr 6, 2023

Ricci flow on Finsler manifolds

Behroz Bidabad, Maral K. Sedaghat

math.DGarXiv:2304.03005
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Abstract

This paper investigates the short-time existence and uniqueness of Ricci flow solutions on Finsler manifolds. The main results of this paper are theorems demonstrating the short-time existence of the flow solution for -dimensional Finsler manifolds and the uniqueness of the solution for isotropic Finsler manifolds. Two examples are also presented to illustrate the results.

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Submitted
Apr 6, 2023
Subjects
math.DG

Comments: 28 pages. arXiv admin note: substantial text overlap with arXiv:1807.04142

Ricci flowStability & uniquenessHarmonic maps & DeTurck trick

Its neighbourhood

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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 Short time existence for harmonic map heat flow with time-dependent metrics (Shaochuang Huang and Luen-Fai Tam, 2021)1Huang 2021 Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion (Helmut Abels, Felicitas Bürger and Harald Garcke, 2022)2Abels 2022 Short time existence and smoothness of the nonlocal mean curvature flow of graphs (Anoumou Attiogbe, Mouahmed Moustapha Fall and Tobias Weth, 2022)6Attiogbe 2022 Bounds of Scalar curvature, S-curvature and distortion on ∞-Einstein Finsler manifolds (Bin Shen, 2024)5Shen 2024 BRIDGES Lectures: Flows of geometric structures, especially G₂-structures (Spiro Karigiannis, 2025)3Karigiannis 2025 Ricci-harmonic flow of G₂ and Spin(7)-structures (Shubham Dwivedi, 2026)7Dwivedi 2026 Smoothing polyhedral spaces via Ricci flow (Richard H. Bamler and Esther Cabezas-Rivas, 2026)4Bamler 2026 Almost Hermitian geometry and generalized Ricci flow (Anna Fino and Julieth Saavedra, 2026)8Fino 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Short time existence for harmonic map heat flow with time-dependent metrics

    Shaochuang Huang and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

  2. Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion

    Helmut Abels, Felicitas Bürger and Harald GarckeWider flows

    Topics Harmonic maps & DeTurck trick · Shared terms “short time”, “time existence”

  3. BRIDGES Lectures: Flows of geometric structures, especially -structures

    Spiro Karigiannis

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

  4. Smoothing polyhedral spaces via Ricci flow

    Richard H. Bamler and Esther Cabezas-Rivas

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

  5. Bounds of Scalar curvature, S-curvature and distortion on -Einstein Finsler manifolds

    Bin Shen

    Shared terms “Finsler manifolds”

  6. Short time existence and smoothness of the nonlocal mean curvature flow of graphs

    Anoumou Attiogbe, Mouahmed Moustapha Fall and Tobias WethWider flows

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

  7. Ricci-harmonic flow of and Spin(7)-structures

    Shubham Dwivedi

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

  8. Almost Hermitian geometry and generalized Ricci flow

    Anna Fino and Julieth Saavedra

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

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.