Paper · Aug 6, 2016

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

Gregory Drugan, Frederick Tsz-Ho Fong, Hojoo Lee

math.DGWider flowsarXiv:1608.02137
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Abstract

We show that any complete, immersed self-expander to the inverse mean curvature flow, which has one end asymptotic to a cylinder, or has two ends asymptotic to two coaxial cylinders, must be rotationally symmetric.

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Submitted
Aug 6, 2016
Subjects
math.DG
Mean curvature flowOther curvature flowsInverse mean curvature flowSelf-shrinkers & translatorsRotational symmetry & warped products

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 Weighted Hsiung-Minkowski formulas and rigidity of umbilical hypersurfaces (Kwok-Kun Kwong, Hojoo Lee and Juncheol Pyo, 2016)7Kwong 2016 Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders (Frederick Tsz-Ho Fong and Peter McGrath, 2016)2Fong 2016 Self-Expanders to Inverse Curvature Flows by Homogeneous Functions (Tsz-Kiu Aaron Chow, Ka-Wing Chow and Frederick Tsz-Ho Fong, 2017)1Chow 2017 A survey of closed self-shrinkers with symmetry (Gregory Drugan, Hojoo Lee and Xuan Hien Nguyen, 2017)3Drugan 2017 Existence of self-similar solution of the inverse mean curvature flow (K. M. Hui, 2018)6Hui 2018 Existence of hypercylinder expanders of the inverse mean curvature flow (K. M. Hui, 2018)8Hui 2018 Self similar solutions to the inverse mean curvature flow in ℝ² (Jui-En Chang, 2018)5Chang 2018 Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces (Frederick Tsz-Ho Fong, 2024)4Fong 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Self-Expanders to Inverse Curvature Flows by Homogeneous Functions

    Tsz-Kiu Aaron Chow, Ka-Wing Chow and Frederick Tsz-Ho FongWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Self-shrinkers & translators · Same author Frederick Tsz-Ho Fong · Shared terms “self expanders”, “cylindrical end”, “rotationally symmetric”

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

    Frederick Tsz-Ho Fong and Peter McGrathWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Same author Frederick Tsz-Ho Fong · Shared terms “self expanders”, “rotationally symmetric”

  3. A survey of closed self-shrinkers with symmetry

    Gregory Drugan, Hojoo Lee and Xuan Hien NguyenWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Same authors Gregory Drugan, Hojoo Lee

  4. Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces

    Frederick Tsz-Ho FongWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Self-shrinkers & translators · Same author Frederick Tsz-Ho Fong · Shared terms “self expanders”, “rotationally symmetric”

  5. Self similar solutions to the inverse mean curvature flow in

    Jui-En ChangWider flows

    Topics Inverse mean curvature flow, Other curvature flows

  6. Existence of self-similar solution of the inverse mean curvature flow

    K. M. HuiWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Other curvature flows

  7. Weighted Hsiung-Minkowski formulas and rigidity of umbilical hypersurfaces

    Kwok-Kun Kwong, Hojoo Lee and Juncheol Pyo

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Self-shrinkers & translators · Same author Hojoo Lee · Shared terms “immersed self”

  8. Existence of hypercylinder expanders of the inverse mean curvature flow

    K. M. Hui

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Other curvature flows · Shared terms “expanders”

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) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) 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.