Paper · Aug 29, 2017

Shrinking doughnuts via variational methods

Gregory Drugan, Xuan Hien Nguyen

math.DGWider flowsarXiv:1708.08808
View PDF View HTML arXiv page

As on arXiv

Abstract

We use variational methods and a modified curvature flow to give an alternative proof of the existence of a self-shrinking torus under mean curvature flow. As a consequence of the proof, we establish an upper bound for the weighted energy of our shrinking doughnuts.

View PDF View HTML arXiv page
Submitted
Aug 29, 2017
Subjects
math.DG

Comments: 18 pages, 2 figures

Mean curvature flowOther curvature flowsSelf-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 self-expanders to the inverse mean curvature flow with cylindrical ends (Gregory Drugan, Frederick Tsz-Ho Fong and Hojoo Lee, 2016)4Drugan 2016 A survey of closed self-shrinkers with symmetry (Gregory Drugan, Hojoo Lee and Xuan Hien Nguyen, 2017)1Drugan 2017 On the Existence of a Closed, Embedded, Rotational λ-Hypersurface (John Ross, 2017)7Ross 2017 On the construction of closed nonconvex nonsoliton ancient mean curvature flows (Theodora Bourni, Mathew Langford and Alexander Mramor, 2019)2Bourni 2019 Compactness and rigidity of self-shrinking surfaces (Tang-Kai Lee, 2021)8Lee 2021 Closed embedded self-shrinkers of mean curvature flow (Oskar Riedler, 2022)5Riedler 2022 Genus one singularities in mean curvature flow (Adrian Chun-Pong Chu and Ao Sun, 2023)3Chu 2023 Embedded cylindrical and doughnut-shaped λ-hypersurfaces (Qing-Ming Cheng, Junqi Lai and Guoxin Wei, 2024)6Cheng 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A survey of closed self-shrinkers with symmetry

    Gregory Drugan, Hojoo Lee and Xuan Hien NguyenWider flows

    Topics Self-shrinkers & translators · Same authors Gregory Drugan, Xuan Hien Nguyen

  2. 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 doughnut”, “shrinking torus”, “self shrinking”

  3. Genus one singularities in mean curvature flow

    Adrian Chun-Pong Chu and Ao SunWider flows

    Topics Self-shrinkers & translators · Shared terms “shrinking doughnut”

  4. Rotational symmetry of self-expanders to the inverse mean curvature flow with cylindrical ends

    Gregory Drugan, Frederick Tsz-Ho Fong and Hojoo LeeWider flows

    Topics Self-shrinkers & translators, Other curvature flows · Same author Gregory Drugan

  5. Closed embedded self-shrinkers of mean curvature flow

    Oskar RiedlerWider flows

    Topics Self-shrinkers & translators · Shared terms “shrinking doughnut”

  6. Embedded cylindrical and doughnut-shaped -hypersurfaces

    Qing-Ming Cheng, Junqi Lai and Guoxin WeiWider flows

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

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

    John RossWider flows

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

  8. Compactness and rigidity of self-shrinking surfaces

    Tang-Kai LeeWider flows

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

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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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.