Paper · Nov 18, 2024

Liouville theorems for harmonic 1-forms on gradient Ricci solitons

Chenghong He, Di Wu, Xi Zhang

math.DGarXiv:2411.12012v2
PDF Abstract page

As on arXiv

Abstract

We prove that there is no nontrivial -integrable harmonic 1-form on noncompact complete gradient steady Ricci solitons or noncompact complete gradient shrinking Kähler-Ricci solitons. As an application, it can be used to distinguish certain flat vector bundles that arise from fundamental group representations into .

Ricci solitonsKähler–Ricci solitonsKähler geometryBoundaries & non-compact

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 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 Liouville Theorem on Ricci shrinkers with constant scalar curvature and its application (Weixiong Mai and Jianyu Ou, 2022)7Mai 2022 L^p Liouville type theorems for harmonic functions on gradient Ricci solitons (Yong Luo, 2023)2Luo 2023 Three circles theorems and Liouville type theorems (Run-Qiang Jian and Zhu-Hong Zhang, 2024)3Jian 2024 A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity (Richard H. Bamler, Eric Chen and Ronan J. Conlon, 2026)6Bamler 2026 Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2026)4Conlon 2026 L^p Liouville theorems for pluriharmonic functions on gradient Kähler-Ricci solitons (Guangwen Zhao, 2026)1Zhao 2026 Kählerity of complete almost-Kähler gradient shrinking Ricci solitons (Junming Xie, 2026)5Xie 2026 Linear Stability of Steady and Expanding Kähler-Ricci Solitons (Lucas Lavoyer and Adam Thompson, 2026)8Lavoyer 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Liouville theorems for pluriharmonic functions on gradient Kähler-Ricci solitons

    Guangwen Zhao PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “Liouville theorem”, “complete gradient”

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

    Yong Luo PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “Liouville”, “gradient Ricci”, “gradient shrinking”

  3. Three circles theorems and Liouville type theorems

    Run-Qiang Jian and Zhu-Hong Zhang PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “Liouville”, “shrinking Kähler”, “gradient shrinking”

  4. Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones

    Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Shared terms “shrinking Kähler”

  5. Kählerity of complete almost-Kähler gradient shrinking Ricci solitons

    Junming Xie PDF

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Shared terms “gradient shrinking”, “complete gradient”, “shrinking Kähler”

  6. A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity

    Richard H. Bamler, Eric Chen and Ronan J. Conlon PDF

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Shared terms “gradient Ricci”

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

    Weixiong Mai and Jianyu Ou PDF

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

  8. Linear Stability of Steady and Expanding Kähler-Ricci Solitons

    Lucas Lavoyer and Adam Thompson PDF

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry

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) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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.