Paper · May 20, 2020

On Ecker's local integral quantity at infinity for ancient mean curvature flows

Keita Kunikawa

math.DGWider flowsarXiv:2005.09845v2
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Abstract

We point out that Ecker's local integral quantity agrees with Huisken's global integral quantity at infinity for ancient mean curvature flows if Huisken's one is finite on each time-slice. In particular, this means that the finiteness of Ecker's integral quantity at infinity implies the finiteness of the entropy at infinity.

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Submitted
May 20, 2020
Last revised
Jun 18, 2020 (v2)
Subjects
math.DG

Comments: 11 pages. Corrected typos. Added some comments in Remark 1.4

Mean curvature flowAncient solutions

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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 Non-existence of eternal solutions to Lagrangian mean curvature flow with non-negative Ricci curvature (Keita Kunikawa, 2016)1Kunikawa 2016 Ancient mean curvature flows and their spacetime tracks (Francesco Chini and Niels Martin Møller, 2019)4Chini 2019 Construction of High Codimension Ancient Mean Curvature Flows (Douglas Stryker and Ao Sun, 2019)5Stryker 2019 Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at -∞ (Douglas Stryker and Ao Sun, 2019)8Stryker 2019 On the construction of closed nonconvex nonsoliton ancient mean curvature flows (Theodora Bourni, Mathew Langford and Alexander Mramor, 2019)2Bourni 2019 Collapsing and noncollapsing in convex ancient mean curvature flow (Theodora Bourni, Mat Langford and Stephen Lynch, 2021)6Bourni 2021 Ancient mean curvature flows from minimal hypersurfaces (Yongheng Han, 2023)3Han 2023 Ancient mean curvature flow asymptotic to a minimal quadratic cone (Junyoung Park, 2026)7Park 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Non-existence of eternal solutions to Lagrangian mean curvature flow with non-negative Ricci curvature

    Keita KunikawaWider flows

    Topics Ancient solutions · Same author Keita Kunikawa

  2. On the construction of closed nonconvex nonsoliton ancient mean curvature flows

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    Topics Ancient solutions

  3. Ancient mean curvature flows from minimal hypersurfaces

    Yongheng HanWider flows

    Topics Ancient solutions

  4. Ancient mean curvature flows and their spacetime tracks

    Francesco Chini and Niels Martin MøllerWider flows

    Topics Ancient solutions

  5. Construction of High Codimension Ancient Mean Curvature Flows

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    Topics Ancient solutions

  6. Collapsing and noncollapsing in convex ancient mean curvature flow

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    Topics Ancient solutions

  7. Ancient mean curvature flow asymptotic to a minimal quadratic cone

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    Topics Ancient solutions

  8. Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at

    Douglas Stryker and Ao SunWider flows

    Topics Ancient solutions

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)

In the guidebook

  1. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.

  2. 12B.1
    κ-SolutionsBook 12B · κ-Solutions and Surgery

    Ancient, nonnegatively curved, noncollapsed: the models for singularities.

  3. 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.