Paper · Jul 11, 2018

Ricci flow on Finsler surfaces

Behroz Bidabad, Maral K. Sedaghat

math.DGarXiv:1807.04142
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Abstract

Here, we study the existence and uniqueness of solutions to the Ricci flow on Finsler surfaces and show short time existence of solutions for such flows. To this purpose, we first study the Finslerian Ricci-DeTurck flow on Finsler surfaces and find a unique short time solution to this flow. Then, we find a solution to the original Ricci flow by pulling back the solution of the Ricci-DeTurck flow using appropriate diffeomorphisms. In the end, we illustrate this argument with some examples.

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Submitted
Jul 11, 2018
Subjects
math.DG
Journal
Journal of Geometry and Physics, Vol. 129, July 2018, Pages 238-254
DOI
10.1016/j.geomphys.2018.02.019

Comments: 32

Ricci flowStability & uniquenessHarmonic maps & DeTurck trick

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 A simple proof of the short-time existence and uniqueness for Ricci flow (Kaveh Eftekharinasab, 2016)6Eftekharinas 2016 Short-Time Existence Theorem for the CR Torsion Flow (Shu-Cheng Chang and Chin-Tung Wu, 2018)4Chang 2018 Complete Ricci solitons on Finsler manifolds (Behroz Bidabad and Mohammad Yar Ahmadi, 2018)8Bidabad 2018 Polyhomogéniété des métriques compatibles avec une structure de Lie à l'infini le long du flot de Ricci (Mahdi Ammar, 2019)5Ammar 2019 ADM mass for C⁰ metrics and distortion under Ricci-DeTurck flow (Paula Burkhardt-Guim, 2022)7Burkhardt-Gu 2022 Ricci flow on Finsler manifolds (Behroz Bidabad and Maral K. Sedaghat, 2023)1Bidabad 2023 Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition (Man-Chun Lee and Stephen Shang Yi Liu, 2024)3Lee 2024 The Ricci-DeTurck flow on complete manifolds (Jing-Bin Cai and Bing Wang, 2026)2Cai 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 Finsler manifolds

    Behroz Bidabad and Maral K. Sedaghat

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Same authors Behroz Bidabad, Maral K. Sedaghat · Shared terms “Finsler”, “short time”, “time existence”

  2. The Ricci-DeTurck flow on complete manifolds

    Jing-Bin Cai and Bing Wang

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

  3. Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition

    Man-Chun Lee and Stephen Shang Yi Liu

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

  4. Short-Time Existence Theorem for the CR Torsion Flow

    Shu-Cheng Chang and Chin-Tung Wu

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

  5. Polyhomogéniété des métriques compatibles avec une structure de Lie à l'infini le long du flot de Ricci

    Mahdi Ammar

    Topics Harmonic maps & DeTurck trick

  6. A simple proof of the short-time existence and uniqueness for Ricci flow

    Kaveh Eftekharinasab

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

  7. ADM mass for metrics and distortion under Ricci-DeTurck flow

    Paula Burkhardt-Guim

    Topics Harmonic maps & DeTurck trick

  8. Complete Ricci solitons on Finsler manifolds

    Behroz Bidabad and Mohammad Yar Ahmadi

    Topics Ricci flow · Same author Behroz Bidabad · Shared terms “Finsler”

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.