Paper · May 11, 2020

Liouville theorem for heat equation along ancient super Ricci flow via reduced geometry

Keita Kunikawa, Yohei Sakurai

math.DGarXiv:2005.04882v2
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Abstract

The aim of this article is to provide a Liouville theorem for heat equation along ancient super Ricci flow. We formulate such a Liouville theorem under a growth condition concerning Perelman's reduced distance.

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Submitted
May 11, 2020
Last revised
Jun 2, 2021 (v2)
Subjects
math.DG

Comments: 23 pages

Ricci flowAncient solutionsReduced volumeWeak & singular flows

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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 2016, 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 Parametric Solution of a Small-Large Black Hole Coexistence Curve (Shanshan Li, Dan-Dan Li, Li-Qin Mi et al., 2017)8Li 2017 Analysis of Type I Singularities in the Harmonic Ricci Flow (Gianmichele Di Matteo, 2018)7Di Matteo 2018 Hamilton type entropy formula along the Ricci flow on surfaces with boundary (Keita Kunikawa and Yohei Sakurai, 2020)5Kunikawa 2020 Liouville theorems for harmonic map heat flow along ancient super Ricci flow via reduced geometry (Keita Kunikawa and Yohei Sakurai, 2021)1Kunikawa 2021 Gaussian heat kernel estimates of Bamler-Zhang type along super Ricci flow (Keita Kunikawa and Yohei Sakurai, 2022)3Kunikawa 2022 Almost splitting and quantitative stratification for super Ricci flow (Keita Kunikawa and Yohei Sakurai, 2023)2Kunikawa 2023 An ε-regularity theorem for Perelman's reduced volume (Liang Cheng and Yongjia Zhang, 2025)4Cheng 2025 On a parabolic curvature lower bound generalizing Ricci flows (Marco Flaim and Erik Hupp, 2025)6Flaim 2025
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 harmonic map heat flow 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”, “concerning”

  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. An -regularity theorem for Perelman's reduced volume

    Liang Cheng and Yongjia Zhang

    Topics Reduced volume · Shared terms “Perelman reduced”

  5. Hamilton type entropy formula along the Ricci flow on surfaces with boundary

    Keita Kunikawa and Yohei Sakurai

    Topics Ricci flow · Same authors Keita Kunikawa, Yohei Sakurai · Shared terms “Perelman”

  6. On a parabolic curvature lower bound generalizing Ricci flows

    Marco Flaim and Erik Hupp

    Topics Reduced volume, Weak & singular flows · Shared terms “super Ricci”, “heat equation”, “Perelman”

  7. Analysis of Type I Singularities in the Harmonic Ricci Flow

    Gianmichele Di Matteo

    Topics Reduced volume

  8. Parametric Solution of a Small-Large Black Hole Coexistence Curve

    Shanshan Li, Dan-Dan Li, Li-Qin Mi et al.

    Topics Reduced volume

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) Reduced volume Perelman's ℒ-length is a space-time distance adapted to Ricci flow. The reduced volume built from it is monotone, playing the role Huisken's… 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)

In the guidebook

  1. 12A.5
    Reduced Distance and Reduced VolumeBook 12A · Entropy and Noncollapsing

    A space-time Bishop–Gromov theorem.

  2. 12A.6
    𝓛-Geodesics and the Variational FormulasBook 12A · Entropy and Noncollapsing

    Perelman's §7.1 derived line by line: the first and second variation of 𝓛-length, 𝓛-Jacobi fields, adapted fields, Hamilton's Harnack expressions, and the differential inequalities for the reduced distance.

  3. 12A.7
    Monotonicity of the Reduced VolumeBook 12A · Entropy and Noncollapsing

    The 𝓛-exponential map and its Jacobian, the injectivity domain, the limit at τ = 0, barrier inequalities on complete flows, and min ℓ ≤ n/2, following Morgan and Tian's chapters 6 and 7.