Paper · Dec 8, 2020

Some characterizations of -Einstein solitons

Absos Ali Shaikh, Antonio W. Cunha, Prosenjit Mandal

math.DGarXiv:2101.03896
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As on arXiv

Abstract

In this article we have showed that a gradient -Einstein soliton with a vector field of bounded norm and satisfying some other conditions is isometric to the Euclidean sphere. Later, we have proved that a non-trivial complete gradient -Einstein soliton with finite weighted Dirichlet integral and certain restriction on Ricci curvature must be of constant scalar curvature and steady Ricci flat. Finally, we have proved that a non-shrinking or non-expanding gradient traceless Ricci soliton possessing some conditions must be steady.

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Submitted
Dec 8, 2020
Subjects
math.DG
DOI
10.1016/j.geomphys.2021.104270
Ricci solitonsRicci-flat & Calabi–YauScalar curvature

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 Some results on η-Ricci Soliton and gradient ρ-Einstein soliton in a complete Riemannian manifold (Absos Ali Shaikh and Chandan Kumar Mondal, 2018)3Shaikh 2018 On Ricci solitons whose potential is convex (Chandan Kumar Mondal and Absos Ali Shaikh, 2019)1Mondal 2019 Curvature estimates for steady Ricci solitons (Pak-Yeung Chan, 2019)6Chan 2019 Isometry theorem of Cartan-Hadamard manifold (Absos Ali Shaikh, Prosenjit Mandal, Chandan Kumar Mondal et al., 2020)4Shaikh 2020 Isometry theorem of gradient Shrinking Ricci solitons (Absos Ali Shaikh and Chandan Kumar Mondal, 2020)7Shaikh 2020 Compact gradient ρ-Einstein soliton is isometric to the Euclidean sphere (Absos Ali Shaikh, Chandan Kumar Mondal and Prosenjit Mandal, 2020)2Shaikh 2020 A comparison theorem for steady Ricci solitons (Benedito Leandro and Jeferson Poveda, 2022)8Leandro 2022 Triviality Results and Conjugate Radius Estimation of Ricci Solitons (Absos Ali Shaikh, Prosenjit Mandal and V. Amarendra Babu, 2023)5Shaikh 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On Ricci solitons whose potential is convex

    Chandan Kumar Mondal and Absos Ali Shaikh

    Topics Ricci-flat & Calabi–Yau, Scalar curvature, Ricci solitons · Same author Absos Ali Shaikh · Shared terms “Dirichlet integral”, “weighted Dirichlet”, “finite weighted”

  2. Compact gradient -Einstein soliton is isometric to the Euclidean sphere

    Absos Ali Shaikh, Chandan Kumar Mondal and Prosenjit Mandal

    Topics Scalar curvature, Ricci solitons · Same authors Absos Ali Shaikh, Prosenjit Mandal · Shared terms “Einstein soliton”, “Euclidean sphere”

  3. Some results on -Ricci Soliton and gradient -Einstein soliton in a complete Riemannian manifold

    Absos Ali Shaikh and Chandan Kumar Mondal

    Topics Scalar curvature, Ricci solitons · Same author Absos Ali Shaikh · Shared terms “Einstein soliton”, “Euclidean sphere”, “vector field”

  4. Isometry theorem of Cartan-Hadamard manifold

    Absos Ali Shaikh, Prosenjit Mandal, Chandan Kumar Mondal et al.

    Topics Scalar curvature, Ricci solitons · Same authors Absos Ali Shaikh, Prosenjit Mandal · Shared terms “Euclidean”

  5. Triviality Results and Conjugate Radius Estimation of Ricci Solitons

    Absos Ali Shaikh, Prosenjit Mandal and V. Amarendra Babu

    Topics Ricci solitons · Same authors Absos Ali Shaikh, Prosenjit Mandal · Shared terms “restriction”, “vector field”

  6. Curvature estimates for steady Ricci solitons

    Pak-Yeung Chan

    Topics Ricci-flat & Calabi–Yau, Scalar curvature, Ricci solitons · Shared terms “steady Ricci”

  7. Isometry theorem of gradient Shrinking Ricci solitons

    Absos Ali Shaikh and Chandan Kumar Mondal

    Topics Scalar curvature, Ricci solitons · Same author Absos Ali Shaikh · Shared terms “Euclidean sphere”

  8. A comparison theorem for steady Ricci solitons

    Benedito Leandro and Jeferson Poveda

    Topics Ricci-flat & Calabi–Yau, Scalar curvature, Ricci solitons · Shared terms “steady 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 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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

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