Paper · Aug 15, 2022

Liouville Theorem on Ricci shrinkers with constant scalar curvature and its application

Weixiong Mai, Jianyu Ou

math.DGarXiv:2208.07101
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Abstract

In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate : any bounded harmonic function is constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension.

Ricci solitonsScalar curvature

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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 2021, 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 L^p Liouville type theorems for harmonic functions on gradient Ricci solitons (Yong Luo, 2023)7Luo 2023 The dimension of polynomial growth holomorphic functions and forms on gradient Kähler Ricci shrinkers (Fei He and Jianyu Ou, 2024)6He 2024 Three circles theorems and Liouville type theorems (Run-Qiang Jian and Zhu-Hong Zhang, 2024)5Jian 2024 Some rigidity results on shrinking gradient Ricci soliton (Jianyu Ou, Yuanyuan Qu and Guoqiang Wu, 2024)4Ou 2024 Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al., 2024)1Li 2024 Liouville theorems for harmonic 1-forms on gradient Ricci solitons (Chenghong He, Di Wu and Xi Zhang, 2024)8He 2024 Kähler–Ricci Shrinkers with Constant Scalar Curvature Are Rigid (Wenqi Li, Jianyu Ou, Guoqiang Wu et al., 2026)3Li 2026 Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set (Fengjiang Li, Wenqi Li, Jianyu Ou et al., 2026)2Li 2026
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  1. Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al. PDF

    Topics Scalar curvature, Ricci solitons · Same author Jianyu Ou · Shared terms “constant scalar”, “gradient shrinking”, “shrinking Ricci”

  2. Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set

    Fengjiang Li, Wenqi Li, Jianyu Ou et al. PDF

    Topics Scalar curvature, Ricci solitons · Same author Jianyu Ou · Shared terms “constant scalar”, “Ricci shrinkers”

  3. Kähler–Ricci Shrinkers with Constant Scalar Curvature Are Rigid

    Wenqi Li, Jianyu Ou, Guoqiang Wu et al. PDF

    Topics Scalar curvature, Ricci solitons · Same author Jianyu Ou · Shared terms “constant scalar”, “Ricci shrinkers”

  4. Some rigidity results on shrinking gradient Ricci soliton

    Jianyu Ou, Yuanyuan Qu and Guoqiang Wu PDF

    Topics Scalar curvature, Ricci solitons · Same author Jianyu Ou · Shared terms “constant scalar”

  5. Three circles theorems and Liouville type theorems

    Run-Qiang Jian and Zhu-Hong Zhang PDF

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

  6. The dimension of polynomial growth holomorphic functions and forms on gradient Kähler Ricci shrinkers

    Fei He and Jianyu Ou PDF

    Topics Ricci solitons · Same author Jianyu Ou · Shared terms “polynomial growth”, “Ricci shrinkers”, “gradient shrinking”

  7. Liouville type theorems for harmonic functions on gradient Ricci solitons

    Yong Luo PDF

    Topics Ricci solitons · Shared terms “harmonic functions”, “Liouville”, “gradient shrinking”

  8. Liouville theorems for harmonic 1-forms on gradient Ricci solitons

    Chenghong He, Di Wu and Xi Zhang PDF

    Topics Ricci solitons · Shared terms “Liouville theorem”, “gradient shrinking”

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.