Paper · Jan 22, 2020

Isometry theorem of Cartan-Hadamard manifold

Absos Ali Shaikh, Prosenjit Mandal, Chandan Kumar Mondal, Pinaki Ranjan Ghosh

math.DGarXiv:2001.07965v2
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Abstract

Cartan-Hadamard manifold is a simply connected Riemannian manifold with non-positive sectional curvature. In this article, we have proved that a Cartan-Hadamard manifold satisfying steady gradient Ricci soliton with the integral condition of potential function is isometric to the Euclidean space. Next we have proved a compactness theorem for gradient shrinking Ricci soliton satisfying some scalar curvature condition. Finally, we have showed that a gradient expanding Ricci soliton with linear volume growth and positive potential function is an Einstein manifold.

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Submitted
Jan 22, 2020
Last revised
Jan 23, 2020 (v2)
Subjects
math.DG

Comments: 10 pages. We highly appreciate valuable comments from the interested researchers

Ricci solitonsEinstein metricsSphere theoremsScalar curvatureLimit spaces & RCD

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 Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds (Felix Schulze, 2018)7Schulze 2018 Some results on η-Ricci Soliton and gradient ρ-Einstein soliton in a complete Riemannian manifold (Absos Ali Shaikh and Chandan Kumar Mondal, 2018)4Shaikh 2018 On Ricci solitons whose potential is convex (Chandan Kumar Mondal and Absos Ali Shaikh, 2019)2Mondal 2019 Isometry theorem of gradient Shrinking Ricci solitons (Absos Ali Shaikh and Chandan Kumar Mondal, 2020)1Shaikh 2020 Compact gradient ρ-Einstein soliton is isometric to the Euclidean sphere (Absos Ali Shaikh, Chandan Kumar Mondal and Prosenjit Mandal, 2020)3Shaikh 2020 Translating solitons over Cartan-Hadamard manifolds (Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen et al., 2020)8Casteras 2020 Some characterizations of ρ-Einstein solitons (Absos Ali Shaikh, Antonio W. Cunha and Prosenjit Mandal, 2020)5Shaikh 2020 Triviality Results and Conjugate Radius Estimation of Ricci Solitons (Absos Ali Shaikh, Prosenjit Mandal and V. Amarendra Babu, 2023)6Shaikh 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Isometry theorem of gradient Shrinking Ricci solitons

    Absos Ali Shaikh and Chandan Kumar Mondal

    Topics Scalar curvature, Ricci solitons · Same authors Absos Ali Shaikh, Chandan Kumar Mondal · Shared terms “isometry theorem”, “linear volume”, “gradient shrinking”

  2. On Ricci solitons whose potential is convex

    Chandan Kumar Mondal and Absos Ali Shaikh

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

  3. 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, Chandan Kumar Mondal · Shared terms “integral condition”, “soliton satisfying”, “Euclidean”

  4. 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 Absos Ali Shaikh, Chandan Kumar Mondal · Shared terms “manifolds satisfying”, “Riemannian manifold”, “Euclidean”

  5. Some characterizations of -Einstein solitons

    Absos Ali Shaikh, Antonio W. Cunha and Prosenjit Mandal

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

  6. 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 “gradient shrinking”, “shrinking Ricci”, “gradient Ricci”

  7. Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds

    Felix SchulzeWider flows

    Topics Sphere theorems · Shared terms “Hadamard manifolds”, “Cartan Hadamard”, “positive sectional”

  8. Translating solitons over Cartan-Hadamard manifolds

    Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen et al.Wider flows

    Shared terms “Hadamard manifolds”, “Cartan Hadamard”

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) 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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988) 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. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  3. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.