Paper · Nov 7, 2017

A Liouville Theorem for Mean Curvature Flow

Kevin Sonnanburg

math.DGWider flowsarXiv:1711.02261
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Abstract

Ancient solutions arise in the study of parabolic blow-ups. If we can categorize ancient solutions, we can better understand blow-up limits. Based on an argument of Giga and Kohn, we give a Liouville-type theorem restricting ancient, type-I, non-collapsing two- dimensional mean curvature flows to either spheres or cylinders.

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Submitted
Nov 7, 2017
Subjects
math.DG

Comments: 14 pages, 2 figures

Mean curvature flowAncient solutionsSingularities(Non)collapsingSurfaces

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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 Blow-up Continuity for Type-I, Mean-Convex Mean Curvature Flow (Kevin Sonnanburg, 2017)4Sonnanburg 2017 A collapsing ancient solution of mean curvature flow in ℝ³ (Theodora Bourni, Mat Langford and Giuseppe Tinaglia, 2017)3Bourni 2017 Ancient solutions in Lagrangian mean curvature flow (Ben Lambert, Jason D. Lotay and Felix Schulze, 2019)5Lambert 2019 Uniqueness of convex ancient solutions to hypersurface flows (Stephen Lynch, 2021)6Lynch 2021 The blowdown of ancient noncollapsed mean curvature flows (Wenkui Du and Robert Haslhofer, 2021)8Du 2021 Collapsing and noncollapsing in convex ancient mean curvature flow (Theodora Bourni, Mat Langford and Stephen Lynch, 2021)7Bourni 2021 A Liouville type theorem for ancient Lagrangian mean curvature flows (Arunima Bhattacharya, Micah Warren and Daniel Weser, 2024)1Bhattacharya 2024 Liouville theorem on p-biharmonic map from gradient Ricci soliton (Xiangzhi Cao, 2026)2Cao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Liouville type theorem for ancient Lagrangian mean curvature flows

    Arunima Bhattacharya, Micah Warren and Daniel WeserWider flows

    Topics Ancient solutions · Shared terms “Liouville”, “type theorem”

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

    Xiangzhi Cao

    Topics Surfaces, Ancient solutions · Shared terms “Liouville theorem”

  3. A collapsing ancient solution of mean curvature flow in

    Theodora Bourni, Mat Langford and Giuseppe TinagliaWider flows

    Topics (Non)collapsing, Ancient solutions

  4. Blow-up Continuity for Type-I, Mean-Convex Mean Curvature Flow

    Kevin SonnanburgWider flows

    Topics (Non)collapsing, Singularities · Same author Kevin Sonnanburg

  5. Ancient solutions in Lagrangian mean curvature flow

    Ben Lambert, Jason D. Lotay and Felix SchulzeWider flows

    Topics Ancient solutions, Singularities · Shared terms “blow ups”

  6. Uniqueness of convex ancient solutions to hypersurface flows

    Stephen LynchWider flows

    Topics (Non)collapsing, Ancient solutions, Singularities

  7. Collapsing and noncollapsing in convex ancient mean curvature flow

    Theodora Bourni, Mat Langford and Stephen LynchWider flows

    Topics (Non)collapsing, Ancient solutions, Singularities

  8. The blowdown of ancient noncollapsed mean curvature flows

    Wenkui Du and Robert HaslhoferWider flows

    Topics (Non)collapsing, Ancient solutions, Singularities

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

In the guidebook

  1. 12B.4
    Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery

    Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

  2. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  3. 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.