Paper · Mar 17, 2026

Liouville theorem on p-biharmonic map from gradient Ricci soliton

Xiangzhi Cao

math.DGarXiv:2603.16541v2
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Abstract

In this paper, we are devoted to obtain some results on p-biharmonic map from gradient Ricci soliton, especially on two dimensional cigar soliton.

Ricci solitonsAncient solutionsSurfacesHarmonic maps & DeTurck trick

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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 O(2)-symmetry of 3D steady gradient Ricci solitons (Yi Lai, 2022)8Lai 2022 On harmonic and biharmonic maps from gradient Ricci solitons (Volker Branding, 2022)1Branding 2022 Liouville Theorem on Ricci shrinkers with constant scalar curvature and its application (Weixiong Mai and Jianyu Ou, 2022)3Mai 2022 A Bernstein theorem of ancient solutions to mean curvature flow (Xiangzhi Cao, 2023)5Cao 2023 A Bernstein theorem for ancient solution to symplectic mean curvature flow (Xiangzhi Cao, 2023)4Cao 2023 Liouville theorems for harmonic 1-forms on gradient Ricci solitons (Chenghong He, Di Wu and Xi Zhang, 2024)2He 2024 Convergence of Ricci flow and long-time existence of Harmonic map heat flow (Kyeongsu Choi and Yi Lai, 2025)7Choi 2025 Monotonicity of the Liouville entropy along the Ricci flow on surfaces (Karen Butt, Alena Erchenko, Tristan Humbert et al., 2025)6Butt 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On harmonic and biharmonic maps from gradient Ricci solitons

    Volker Branding PDF

    Topics Harmonic maps & DeTurck trick, Surfaces, Ancient solutions · Shared terms “biharmonic maps”, “dimensional cigar”, “gradient Ricci”

  2. 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 Ricci”

  3. 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”

  4. A Bernstein theorem for ancient solution to symplectic mean curvature flow

    Xiangzhi CaoWider flows PDF

    Topics Ancient solutions · Same author Xiangzhi Cao

  5. A Bernstein theorem of ancient solutions to mean curvature flow

    Xiangzhi CaoWider flows PDF

    Topics Ancient solutions · Same author Xiangzhi Cao

  6. Monotonicity of the Liouville entropy along the Ricci flow on surfaces

    Karen Butt, Alena Erchenko, Tristan Humbert et al. PDF

    Topics Surfaces · Shared terms “Liouville”

  7. Convergence of Ricci flow and long-time existence of Harmonic map heat flow

    Kyeongsu Choi and Yi Lai PDF

    Topics Harmonic maps & DeTurck trick, Ancient solutions, Ricci solitons

  8. O(2)-symmetry of 3D steady gradient Ricci solitons

    Yi Lai PDF

    Topics Ancient solutions, Ricci solitons · Shared terms “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) 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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.