Paper · Aug 22, 2019

On Ricci solitons whose potential is convex

Chandan Kumar Mondal, Absos Ali Shaikh

math.DGarXiv:1908.08303v3
View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper we consider the Ricci curvature of a Ricci soliton. In particular, we have showed that a complete gradient Ricci soliton with non-negative Ricci curvature possessing a non-constant convex potential function having finite weighted Dirichlet integral satisfying an integral condition is Ricci flat and also it isometrically splits a line. We have also proved that a gradient Ricci soliton with non-constant concave potential function and bounded Ricci curvature is non-shrinking and hence the scalar curvature has at most one critical point.

View PDF View HTML arXiv page
Submitted
Aug 22, 2019
Last revised
Apr 2, 2020 (v3)
Subjects
math.DG

Comments: 8 pages

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)5Shaikh 2018 Curvature estimates for steady Ricci solitons (Pak-Yeung Chan, 2019)7Chan 2019 Isometry theorem of Cartan-Hadamard manifold (Absos Ali Shaikh, Prosenjit Mandal, Chandan Kumar Mondal et al., 2020)3Shaikh 2020 Isometry theorem of gradient Shrinking Ricci solitons (Absos Ali Shaikh and Chandan Kumar Mondal, 2020)2Shaikh 2020 Compact gradient ρ-Einstein soliton is isometric to the Euclidean sphere (Absos Ali Shaikh, Chandan Kumar Mondal and Prosenjit Mandal, 2020)4Shaikh 2020 Some characterizations of ρ-Einstein solitons (Absos Ali Shaikh, Antonio W. Cunha and Prosenjit Mandal, 2020)1Shaikh 2020 A comparison theorem for steady Ricci solitons (Benedito Leandro and Jeferson Poveda, 2022)8Leandro 2022 Conjugate radius and parabolicity along the Ricci flow (Chandan Kumar Mondal, 2026)6Mondal 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Some characterizations of -Einstein solitons

    Absos Ali Shaikh, Antonio W. Cunha and Prosenjit Mandal

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

  2. Isometry theorem of gradient Shrinking Ricci solitons

    Absos Ali Shaikh and Chandan Kumar Mondal

    Topics Scalar curvature, Ricci solitons · Same authors Chandan Kumar Mondal, Absos Ali Shaikh · Shared terms “potential function”

  3. Isometry theorem of Cartan-Hadamard manifold

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

    Topics Scalar curvature, Ricci solitons · Same authors Chandan Kumar Mondal, Absos Ali Shaikh · Shared terms “potential function”, “integral condition”, “gradient Ricci”

  4. 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 Chandan Kumar Mondal, Absos Ali Shaikh · Shared terms “integral condition”

  5. 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 authors Chandan Kumar Mondal, Absos Ali Shaikh

  6. Conjugate radius and parabolicity along the Ricci flow

    Chandan Kumar Mondal

    Topics Ricci-flat & Calabi–Yau, Scalar curvature, Ricci solitons · Same author Chandan Kumar Mondal

  7. Curvature estimates for steady Ricci solitons

    Pak-Yeung Chan

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

  8. A comparison theorem for steady Ricci solitons

    Benedito Leandro and Jeferson Poveda

    Topics Ricci-flat & Calabi–Yau, Scalar curvature, Ricci solitons · Shared terms “potential function”, “gradient 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.