Paper · Sep 15, 2017

On the Existence of a Closed, Embedded, Rotational -Hypersurface

John Ross

math.DGWider flowsarXiv:1709.05020
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Abstract

In this paper we show the existence of a closed, embedded -hypersurfaces . The hypersurface is diffeomorhic to and exhibits symmetry. Our approach uses a "shooting method" similar to the approach used by McGrath in constructing a generalized self-shrinking torus solution to mean curvature flow. The result generalizes the -torus found by Cheng and Wei.

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Submitted
Sep 15, 2017
Subjects
math.DG

Comments: 11 pages

Mean curvature flowSelf-shrinkers & translators

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 Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders (Frederick Tsz-Ho Fong and Peter McGrath, 2016)8Fong 2016 Shrinking doughnuts via variational methods (Gregory Drugan and Xuan Hien Nguyen, 2017)4Drugan 2017 A survey of closed self-shrinkers with symmetry (Gregory Drugan, Hojoo Lee and Xuan Hien Nguyen, 2017)5Drugan 2017 On the construction of closed nonconvex nonsoliton ancient mean curvature flows (Theodora Bourni, Mathew Langford and Alexander Mramor, 2019)1Bourni 2019 Ruled surfaces of generalized self-similar solutions of the mean curvature flow (Rafael López, 2020)3López 2020 Surfaces with prescribed mean curvature in ℍ²×ℝ (Antonio Bueno and Irene Ortiz, 2020)6Bueno 2020 Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions (Jingze Zhu, 2021)2Zhu 2021 Closed embedded self-shrinkers of mean curvature flow (Oskar Riedler, 2022)7Riedler 2022
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 Self-shrinkers & translators · Shared terms “shrinking torus”, “self shrinking”

  2. Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions

    Jingze ZhuWider flows

    Topics Self-shrinkers & translators · Shared terms “rotational”

  3. Ruled surfaces of generalized self-similar solutions of the mean curvature flow

    Rafael LópezWider flows

    Topics Self-shrinkers & translators · Shared terms “generalized self”

  4. Shrinking doughnuts via variational methods

    Gregory Drugan and Xuan Hien NguyenWider flows

    Topics Self-shrinkers & translators · Shared terms “shrinking torus”, “self shrinking”

  5. A survey of closed self-shrinkers with symmetry

    Gregory Drugan, Hojoo Lee and Xuan Hien NguyenWider flows

    Topics Self-shrinkers & translators · Shared terms “rotational”

  6. Surfaces with prescribed mean curvature in

    Antonio Bueno and Irene OrtizWider flows

    Topics Self-shrinkers & translators · Shared terms “rotational”

  7. Closed embedded self-shrinkers of mean curvature flow

    Oskar RiedlerWider flows

    Topics Self-shrinkers & translators

  8. Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders

    Frederick Tsz-Ho Fong and Peter McGrathWider flows

    Topics Self-shrinkers & translators · Shared terms “rotational”

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) 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)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.