Paper · Feb 6, 2020

Isometry theorem of gradient Shrinking Ricci solitons

Absos Ali Shaikh, Chandan Kumar Mondal

math.DGarXiv:2002.02216v2
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Abstract

In this paper, we have proved that if a complete conformally flat gradient shrinking Ricci soliton has linear volume growth or the scalar curvature is finitely integrable and also the reciprocal of the potential function is subharmonic, then the manifold is isometric to the Euclidean sphere. As a consequence, we have showed that a four dimensional gradient shrinking Ricci soliton satisfying some conditions is isometric to or or . We have also deduced a condition for the shrinking Ricci soliton to be compact with quadratic volume growth.

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Submitted
Feb 6, 2020
Last revised
May 23, 2020 (v2)
Subjects
math.DG
DOI
10.1016/j.geomphys.2021.104110

Comments: 10 pages

Ricci solitons4-manifoldsScalar 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 Structure at infinity for shrinking Ricci solitons (Ovidiu Munteanu and Jiaping Wang, 2016)7Munteanu 2016 Rigidity of positively curved shrinking Ricci solitons in dimension four (Giovanni Catino, 2018)8Catino 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)5Mondal 2019 Isometry theorem of Cartan-Hadamard manifold (Absos Ali Shaikh, Prosenjit Mandal, Chandan Kumar Mondal et al., 2020)1Shaikh 2020 Compact gradient ρ-Einstein soliton is isometric to the Euclidean sphere (Absos Ali Shaikh, Chandan Kumar Mondal and Prosenjit Mandal, 2020)2Shaikh 2020 Four-dimensional complete gradient shrinking Ricci solitons (Huai-Dong Cao, Ernani Ribeiro and Detang Zhou, 2020)3Cao 2020 Rigidity of four-dimensional Gradient shrinking Ricci solitons (Xu Cheng and Detang Zhou, 2021)6Cheng 2021
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 Cartan-Hadamard manifold

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

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

  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, Chandan Kumar Mondal · Shared terms “Euclidean sphere”, “soliton satisfying”

  3. Four-dimensional complete gradient shrinking Ricci solitons

    Huai-Dong Cao, Ernani Ribeiro and Detang Zhou

    Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “gradient shrinking”, “shrinking Ricci”, “four dimensional”

  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 “Euclidean sphere”

  5. 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”

  6. Rigidity of four-dimensional Gradient shrinking Ricci solitons

    Xu Cheng and Detang Zhou

    Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “gradient shrinking”, “shrinking Ricci”, “dimensional gradient”

  7. Structure at infinity for shrinking Ricci solitons

    Ovidiu Munteanu and Jiaping Wang

    Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “shrinking Ricci”, “gradient shrinking”, “dimensional gradient”

  8. Rigidity of positively curved shrinking Ricci solitons in dimension four

    Giovanni Catino

    Topics 4-manifolds, Scalar curvature, Ricci solitons · Shared terms “shrinking Ricci”, “soliton satisfying”, “four dimensional”

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) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 1982) 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.