Paper · Sep 29, 2016

Liouville theorems, Volume growth, and volume comparison for Ricci shrinkers

Li Ma

math.DGarXiv:1609.09332
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Abstract

In this paper, we study volume growth, Liouville theorem and the local gradient estimate for -harmonic functions, and volume comparison property of unit balls in complete noncompact gradient Ricci shrinkers. We also study integral properties of f-harmonic functions and harmonic functions on such manifolds.

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Submitted
Sep 29, 2016
Subjects
math.DG, math.AP
Journal
Pacific J. Math. 296 (2018) 357-369
DOI
10.2140/pjm.2018.296.357

Comments: 14 pages

Ricci solitonsBoundaries & non-compact

Its neighbourhood

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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 Liouville-type theorems on the complete gradient shrinking Ricci solitons (Huabin Ge and Shijin Zhang, 2016)6Ge 2016 Volume growth of complete submanifolds in gradient Ricci Solitons with bounded weighted mean curvature (Xu Cheng, Matheus Vieira and Detang Zhou, 2019)7Cheng 2019 Volume growth estimates for Ricci solitons and quasi-Einstein manifolds (Xu Cheng, Ernani Ribeiro and Detang Zhou, 2020)3Cheng 2020 Liouville Theorem on Ricci shrinkers with constant scalar curvature and its application (Weixiong Mai and Jianyu Ou, 2022)1Mai 2022 Rigidity and ε-regularity theorems of Ricci shrinkers (Jie Wang and Youde Wang, 2023)8Wang 2023 L^p Liouville type theorems for harmonic functions on gradient Ricci solitons (Yong Luo, 2023)4Luo 2023 Liouville theorems for harmonic 1-forms on gradient Ricci solitons (Chenghong He, Di Wu and Xi Zhang, 2024)2He 2024 Drift-harmonic functions with polynomial growth on asymptotically paraboloidal manifolds (Michael B. Law, 2025)5Law 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Liouville Theorem on Ricci shrinkers with constant scalar curvature and its application

    Weixiong Mai and Jianyu Ou

    Topics Ricci solitons · Shared terms “Liouville theorem”, “harmonic functions”, “Ricci shrinkers”

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

    Chenghong He, Di Wu and Xi Zhang

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “Liouville theorem”, “gradient Ricci”

  3. Volume growth estimates for Ricci solitons and quasi-Einstein manifolds

    Xu Cheng, Ernani Ribeiro and Detang Zhou

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “volume growth”, “noncompact gradient”, “complete noncompact”

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

    Yong Luo

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

  5. Drift-harmonic functions with polynomial growth on asymptotically paraboloidal manifolds

    Michael B. Law

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “harmonic functions”, “gradient Ricci”

  6. Liouville-type theorems on the complete gradient shrinking Ricci solitons

    Huabin Ge and Shijin Zhang

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

  7. Volume growth of complete submanifolds in gradient Ricci Solitons with bounded weighted mean curvature

    Xu Cheng, Matheus Vieira and Detang Zhou

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “volume growth”, “complete noncompact”, “gradient Ricci”

  8. Rigidity and ε-regularity theorems of Ricci shrinkers

    Jie Wang and Youde Wang

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “Ricci shrinkers”

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.