Paper · Apr 17, 2019

Ancient and Eternal Solutions to Mean Curvature Flow from Minimal Surfaces

Alexander Mramor, Alec Payne

math.DGWider flowsarXiv:1904.08439v2
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Abstract

We construct embedded ancient solutions to mean curvature flow related to certain classes of unstable minimal hypersurfaces in for . These provide examples of mean convex yet nonconvex ancient solutions that are not solitons, meaning that they do not evolve by rigid motions or homotheties. Moreover, we construct embedded eternal solutions to mean curvature flow in for . These eternal solutions are not solitons, are -invariant, and are mean convex yet nonconvex. They flow out of the catenoid and are the rotation of a profile curve which becomes infinitely far from the axis of rotation. As , the profile curves converge to a grim reaper for and become flat for . Concerning these eternal solutions, we also show they are asymptotically unique up to scale among the embedded -invariant, eternal solutions with uniformly bounded curvature and a sign on mean curvature.

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Submitted
Apr 17, 2019
Last revised
May 1, 2019 (v2)
Subjects
math.DG, math.AP

Comments: We generalized the result for ancient solutions to all dimensions and fixed minor errors. 26 pages, comments welcome!

Mean curvature flowAncient solutionsSelf-shrinkers & translatorsStability & uniquenessMinimal & CMC surfaces

Its neighbourhood

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 Non-existence of eternal solutions to Lagrangian mean curvature flow with non-negative Ricci curvature (Keita Kunikawa, 2016)5Kunikawa 2016 Nonconvex Surfaces which Flow to Round Points (Alexander Mramor and Alec Payne, 2019)4Mramor 2019 On the construction of closed nonconvex nonsoliton ancient mean curvature flows (Theodora Bourni, Mathew Langford and Alexander Mramor, 2019)1Bourni 2019 Ancient solutions to mean curvature flow for isoparametric submanifolds (Xiaobo Liu and Chuu-Lian Terng, 2019)7Liu 2019 Nonplanar ancient curve shortening flows in ℝ³ from grim reapers (Theodora Bourni and Alexander Mramor, 2022)2Bourni 2022 Nonconvex ancient solutions to Curve Shortening Flow (Yongzhe Zhang, Connor Olson, Ilyas Khan et al., 2022)8Zhang 2022 Ancient mean curvature flows from minimal hypersurfaces (Yongheng Han, 2023)3Han 2023 On the long-time limit of the mean curvature flow in closed manifolds (Alexander Mramor and Ao Sun, 2024)6Mramor 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the construction of closed nonconvex nonsoliton ancient mean curvature flows

    Theodora Bourni, Mathew Langford and Alexander MramorWider flows

    Topics Ancient solutions, Self-shrinkers & translators · Same author Alexander Mramor · Shared terms “nonconvex”, “mean convex”

  2. Nonplanar ancient curve shortening flows in from grim reapers

    Theodora Bourni and Alexander MramorWider flows

    Topics Ancient solutions, Self-shrinkers & translators · Same author Alexander Mramor · Shared terms “grim reaper”

  3. Ancient mean curvature flows from minimal hypersurfaces

    Yongheng HanWider flows

    Topics Minimal & CMC surfaces, Ancient solutions, Stability & uniqueness · Shared terms “minimal hypersurfaces”, “unstable minimal”, “embedded ancient”

  4. Nonconvex Surfaces which Flow to Round Points

    Alexander Mramor and Alec PayneWider flows

    Topics Mean curvature flow · Same authors Alexander Mramor, Alec Payne · Shared terms “nonconvex”, “mean convex”, “uniformly bounded”

  5. Non-existence of eternal solutions to Lagrangian mean curvature flow with non-negative Ricci curvature

    Keita KunikawaWider flows

    Topics Ancient solutions, Self-shrinkers & translators · Shared terms “eternal solutions”

  6. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows

    Topics Minimal & CMC surfaces · Same author Alexander Mramor

  7. Ancient solutions to mean curvature flow for isoparametric submanifolds

    Xiaobo Liu and Chuu-Lian TerngWider flows

    Topics Minimal & CMC surfaces, Ancient solutions, Stability & uniqueness · Shared terms “minimal hypersurfaces”

  8. Nonconvex ancient solutions to Curve Shortening Flow

    Yongzhe Zhang, Connor Olson, Ilyas Khan et al.Wider flows

    Topics Ancient solutions, Self-shrinkers & translators · Shared terms “nonconvex ancient”, “grim reaper”

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.