Paper · Apr 12, 2021

Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometry

Keita Kunikawa, Yohei Sakurai

math.DGarXiv:2104.05191v2
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Abstract

We study harmonic map heat flow along ancient super Ricci flow, and derive several Liouville theorems with controlled growth from Perelman's reduced geometric viewpoint. For non-positively curved target spaces, our growth condition is sharp. For positively curved target spaces, our Liouville theorem is new even in the static case (i.e., for harmonic maps); moreover, we point out that the growth condition can be improved, and almost sharp in the static case. This fills the gap between the Liouville theorem of Choi and the example constructed by Schoen-Uhlenbeck.

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Submitted
Apr 12, 2021
Last revised
Aug 3, 2021 (v2)
Subjects
math.DG

Comments: 22 pages. arXiv admin note: text overlap with arXiv:2005.04882

Ricci flowAncient solutionsWeak & singular flowsHarmonic maps & DeTurck trick

Its neighbourhood

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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 2016, plus the canon.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Liouville theorem for heat equation along ancient super Ricci flow via reduced geometry

    Keita Kunikawa and Yohei Sakurai

    Topics Ancient solutions, Weak & singular flows · Same authors Keita Kunikawa, Yohei Sakurai · Shared terms “ancient super”, “along ancient”, “reduced geometry”

  2. Almost splitting and quantitative stratification for super Ricci flow

    Keita Kunikawa and Yohei Sakurai

    Topics Weak & singular flows · Same authors Keita Kunikawa, Yohei Sakurai · Shared terms “super Ricci”

  3. Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow

    Keita Kunikawa and Yohei Sakurai

    Topics Weak & singular flows · Same authors Keita Kunikawa, Yohei Sakurai · Shared terms “super Ricci”

  4. Rigidity of Lipschitz map using harmonic map heat flow

    Man-Chun Lee and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick, Weak & singular flows · Shared terms “map heat”, “heat flow”

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

    Shaochuang Huang and Luen-Fai Tam

    Topics Harmonic maps & DeTurck trick · Shared terms “map heat”, “heat flow”, “flow along”

  6. 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 · Shared terms “map heat”, “heat flow”

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

    Xiangzhi Cao

    Topics Harmonic maps & DeTurck trick, Ancient solutions · Shared terms “Liouville theorem”

  8. On the stability of harmonic maps under the homogeneous Ricci flow

    Rafaela F. do Prado and Lino Grama

    Topics Harmonic maps & DeTurck trick

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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) 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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  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.