Paper · Jun 2, 2019

A gap theorem on complete shrinking gradient Ricci solitons

Shijin Zhang

math.DGarXiv:1906.00444
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Abstract

In this short note, using Günther's volume comparison theorem and Yokota's gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton with sectional curvature and for some uniform constant , there exists a small uniform constant depends only on and , if the scalar curvature , then is isometric to the Gaussian soliton .

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Submitted
Jun 2, 2019
Subjects
math.DG
Journal
Proc. Amer. Math. Soc. 146(2018), 359-368
Ricci solitonsScalar curvature

Its neighbourhood

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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 Gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature (Shijin Zhang, 2019)3Zhang 2019 Remarks on shrinking gradient Kähler-Ricci solitons with positive bisectional curvature (Guoqiang Wu and Shijin Zhang, 2019)6Wu 2019 Curvature estimates and gap theorems for expanding Ricci solitons (Pak-Yeung Chan, 2020)5Chan 2020 A new complete two-dimensional shrinking gradient Kähler-Ricci soliton (Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al., 2022)4Bamler 2022 Some rigidity results on shrinking gradient Ricci soliton (Jianyu Ou, Yuanyuan Qu and Guoqiang Wu, 2024)2Ou 2024 Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons (Yukai Sun and Guangrui Zhu, 2025)7Sun 2025 A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Chen Wang and Guoqiang Wu, 2026)8Wang 2026 Rigidity of shrinking gradient ricci soliton with constant scalar curvature (Fengjiang Li, Yuanyuan Qu and Guoqiang Wu, 2026)1Li 2026
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  1. Rigidity of shrinking gradient ricci soliton with constant scalar curvature

    Fengjiang Li, Yuanyuan Qu and Guoqiang Wu

    Topics Scalar curvature, Ricci solitons · Shared terms “shrinking gradient”, “complete shrinking”, “gradient Ricci”

  2. Some rigidity results on shrinking gradient Ricci soliton

    Jianyu Ou, Yuanyuan Qu and Guoqiang Wu

    Topics Scalar curvature, Ricci solitons · Shared terms “shrinking gradient”, “complete shrinking”, “gradient Ricci”

  3. Gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature

    Shijin Zhang

    Topics Ricci solitons · Same author Shijin Zhang · Shared terms “complete shrinking”, “shrinking gradient”

  4. A new complete two-dimensional shrinking gradient Kähler-Ricci soliton

    Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al.

    Topics Scalar curvature, Ricci solitons · Shared terms “shrinking gradient”, “complete shrinking”

  5. Curvature estimates and gap theorems for expanding Ricci solitons

    Pak-Yeung Chan

    Topics Scalar curvature, Ricci solitons · Shared terms “gap theorem”, “gradient Ricci”

  6. Remarks on shrinking gradient Kähler-Ricci solitons with positive bisectional curvature

    Guoqiang Wu and Shijin Zhang

    Topics Ricci solitons · Same author Shijin Zhang · Shared terms “shrinking gradient”, “short note”

  7. Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons

    Yukai Sun and Guangrui Zhu

    Topics Scalar curvature, Ricci solitons · Shared terms “shrinking gradient”, “gradient Ricci”

  8. A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Chen Wang and Guoqiang Wu

    Topics Scalar curvature, Ricci solitons · Shared terms “shrinking gradient”, “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) 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.