Paper · Jul 11, 2018

Complete Ricci solitons on Finsler manifolds

Behroz Bidabad, Mohammad Yar Ahmadi

math.DGarXiv:1807.04128
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Abstract

The geometric flow theory and its applications turned into one of the most intensively developing branches of modern geometry. Here, a brief introduction to Finslerian Ricci flow and their self-similar solutions known as Ricci solitons are given and some recent results are presented. They are a generalization of Einstein metrics and are previously developed by the present authors for Finsler manifolds. In the present work, it is shown that a complete shrinking Ricci soliton Finsler manifold has a finite fundamental group.

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Submitted
Jul 11, 2018
Subjects
math.DG
Journal
Science China Mathematics, 2018

Comments: 14

Ricci flowRicci solitonsEinstein metrics

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 ε-regularity for shrinking Ricci solitons and Ricci flows (Huabin Ge and Wenshuai Jiang, 2017)8Ge 2017 Ricci flow on Finsler surfaces (Behroz Bidabad and Maral K. Sedaghat, 2018)2Bidabad 2018 Harnack inequalities for positive solutions of the heat equation on closed Finsler manifolds (Cyrille Combete, Serge Degla and Leonard Todjihounde, 2018)5Combete 2018 Finslerian geodesics on Fréchet manifolds (Kaveh Eftekharinasab and Valentyna Petrusenko, 2020)4Eftekharinas 2020 Ricci flow on Finsler manifolds (Behroz Bidabad and Maral K. Sedaghat, 2023)1Bidabad 2023 Li-Yau Estimates for a Nonlinear Parabolic Equation on Finsler Manifolds (Bin Shen and Yuhan Zhu, 2023)6Shen 2023 Almost Ricci solitons on Finsler spaces (Qiaoling Xia, 2024)3Xia 2024 Bounds of Scalar curvature, S-curvature and distortion on ∞-Einstein Finsler manifolds (Bin Shen, 2024)7Shen 2024
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 Ricci flow · Same author Behroz Bidabad · Shared terms “Finsler manifolds”, “presented”

  2. Ricci flow on Finsler surfaces

    Behroz Bidabad and Maral K. Sedaghat

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

  3. Almost Ricci solitons on Finsler spaces

    Qiaoling Xia

    Topics Einstein metrics, Ricci solitons · Shared terms “Finsler”

  4. Finslerian geodesics on Fréchet manifolds

    Kaveh Eftekharinasab and Valentyna Petrusenko

    Topics Einstein metrics · Shared terms “Finslerian”

  5. Harnack inequalities for positive solutions of the heat equation on closed Finsler manifolds

    Cyrille Combete, Serge Degla and Leonard Todjihounde

    Topics Ricci flow · Shared terms “Finsler manifolds”

  6. Li-Yau Estimates for a Nonlinear Parabolic Equation on Finsler Manifolds

    Bin Shen and Yuhan Zhu

    Topics Ricci solitons · Shared terms “Finsler manifolds”, “Finslerian”, “developed”

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

    Bin Shen

    Topics Ricci solitons · Shared terms “Finsler manifolds”

  8. -regularity for shrinking Ricci solitons and Ricci flows

    Huabin Ge and Wenshuai Jiang

    Topics Einstein metrics, Ricci solitons · Shared terms “shrinking Ricci”

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  3. 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.