Paper · May 19, 2022

On harmonic and biharmonic maps from gradient Ricci solitons

Volker Branding

math.DGarXiv:2205.09544v2
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Abstract

We study harmonic and biharmonic maps from gradient Ricci solitons. We derive a number of analytic and geometric conditions under which harmonic maps are constant and which force biharmonic maps to be harmonic. In particular, we show that biharmonic maps of finite energy from the two-dimensional cigar soliton must be harmonic.

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Submitted
May 19, 2022
Last revised
Jul 15, 2024 (v2)
Subjects
math.DG
Journal
Math. Nachr. 296 (2023), no. 11, 5109-5122
DOI
10.1002/mana.202200232
Ricci solitonsAncient solutionsSurfacesHarmonic maps & DeTurck trick

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  1. Liouville theorem on p-biharmonic map from gradient Ricci soliton

    Xiangzhi Cao

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

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

    Kyeongsu Choi and Yi Lai

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

  3. Ricci Solitons on the Lie Group and Applications

    Mohammed El Amine Mekki and Ahmed Mohammed Cherif

    Topics Harmonic maps & DeTurck trick, Ricci solitons

  4. Short time existence for harmonic map heat flow with time-dependent metrics

    Shaochuang Huang and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick

  5. On Ricci Solitons and Harmonic Vector Fields in the Thurston Geometry

    Halima Boukhari, Hadjer Okbani and Ahmed Mohammed Cherif

    Topics Harmonic maps & DeTurck trick, Ricci solitons

  6. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick

  7. Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow

    José N. V. Gomes, Matheus Hudson and Carlos M. de Sousa

    Topics Harmonic maps & DeTurck trick, Ricci solitons

  8. Notes on harmonic-Ricci flow on surface

    Xiang-Zhi Cao

    Topics Surfaces

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.