Paper · Nov 7, 2019

Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons

Jia-Yong Wu, Peng Wu

math.DGarXiv:1911.02729v6
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Abstract

In this paper, we study harmonic and caloric functions of polynomial growth on a complete non-compact gradient shrinking Ricci soliton. On one hand, when the scalar curvature satisfies at least quadratic decay, we prove that the space of harmonic functions with fixed polynomial growth degree is finite dimensional. We also prove analogous results for ancient caloric functions. On the other hand, without any curvature condition, we prove sharp finite dimensional estimates for the space of Schrödinger functions with fixed polynomial growth degree.

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Submitted
Nov 7, 2019
Last revised
Dec 24, 2021 (v6)
Subjects
math.DG
Journal
Geom. Dedicata, (2023) 217, Art. 75, 2023

Comments: Theorems 3.2 and 1.2 corrected

Ricci solitonsAncient solutionsScalar curvatureBoundaries & 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 Time analyticity for heat equation on gradient shrinking Ricci solitons (Jia-Yong Wu, 2019)1Wu 2019 Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons (Jia-Yong Wu, 2020)7Wu 2020 Sharp upper diameter bounds for compact shrinking Ricci solitons (Jia-Yong Wu, 2020)5Wu 2020 Geometric inequalities and rigidity of gradient shrinking Ricci solitons (Jia-Yong Wu, 2020)6Wu 2020 Rigidity of four-dimensional Gradient shrinking Ricci solitons (Xu Cheng and Detang Zhou, 2021)8Cheng 2021 Liouville Theorem on Ricci shrinkers with constant scalar curvature and its application (Weixiong Mai and Jianyu Ou, 2022)3Mai 2022 Drift-harmonic functions with polynomial growth on asymptotically paraboloidal manifolds (Michael B. Law, 2025)4Law 2025 Finite generation of the ring of holomorphic functions with polynomial growth on the Kähler-Ricci shrinker (Jiangtao Li, 2025)2Li 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Time analyticity for heat equation on gradient shrinking Ricci solitons

    Jia-Yong Wu

    Topics Boundaries & non-compact, Ricci solitons · Same author Jia-Yong Wu · Shared terms “gradient shrinking”, “shrinking Ricci”, “compact gradient”

  2. Finite generation of the ring of holomorphic functions with polynomial growth on the Kähler-Ricci shrinker

    Jiangtao Li

    Topics Scalar curvature, Boundaries & non-compact, Ricci solitons · Shared terms “polynomial growth”, “gradient shrinking”, “compact gradient”

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

    Weixiong Mai and Jianyu Ou

    Topics Scalar curvature, Ricci solitons · Shared terms “polynomial growth”, “harmonic functions”, “gradient shrinking”

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

    Michael B. Law

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

  5. Sharp upper diameter bounds for compact shrinking Ricci solitons

    Jia-Yong Wu

    Topics Scalar curvature, Ricci solitons · Same author Jia-Yong Wu · Shared terms “shrinking Ricci”, “gradient shrinking”

  6. Geometric inequalities and rigidity of gradient shrinking Ricci solitons

    Jia-Yong Wu

    Topics Ricci solitons · Same author Jia-Yong Wu · Shared terms “shrinking Ricci”, “gradient shrinking”, “Schrödinger”

  7. Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons

    Jia-Yong Wu

    Topics Ricci solitons · Same author Jia-Yong Wu · Shared terms “Schrödinger”, “gradient shrinking”, “shrinking Ricci”

  8. Rigidity of four-dimensional Gradient shrinking Ricci solitons

    Xu Cheng and Detang Zhou

    Topics Scalar curvature, Boundaries & non-compact, Ricci solitons · Shared terms “gradient shrinking”, “shrinking 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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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) 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. 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.

  2. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  3. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.