Paper · Jan 15, 2017

Self-Expanders to Inverse Curvature Flows by Homogeneous Functions

Tsz-Kiu Aaron Chow, Ka-Wing Chow, Frederick Tsz-Ho Fong

math.DGWider flowsarXiv:1701.03995v2
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Abstract

In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-expanders to any of these flows are round spheres. Secondly, we show that complete non-compact self-expanders to any of these flows with asymptotically cylindrical ends must be rotationally symmetric. Thirdly, we show that when such a flow is uniformly parabolic, there exist complete rotationally symmetric self-expanders which are asymptotic to two round cylinders with different radii. These extend some earlier results of inverse mean curvature flow to a wider class of flows.

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Submitted
Jan 15, 2017
Last revised
Jun 16, 2018 (v2)
Subjects
math.DG, math.AP

Comments: to appear in Comm. Anal. Geom

Mean curvature flowOther curvature flowsInverse mean curvature flowSelf-shrinkers & translatorsHomogeneous spacesBoundaries & non-compactRotational 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 Rotational symmetry of self-expanders to the inverse mean curvature flow with cylindrical ends (Gregory Drugan, Frederick Tsz-Ho Fong and Hojoo Lee, 2016)1Drugan 2016 Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders (Frederick Tsz-Ho Fong and Peter McGrath, 2016)4Fong 2016 Existence of self-shrinkers to the degree-one curvature flow with a rotationally symmetric conical end (Siao-Hao Guo, 2016)6Guo 2016 Existence of self-similar solution 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 Theorems of Self-Conformal Solutions to Inverse Curvature Flows (Nicholas Cheng-Hoong Chin, Frederick Tsz-Ho Fong and Jingbo Wan, 2018)3Chin 2018 Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces (Frederick Tsz-Ho Fong, 2024)2Fong 2024 Self-expanders of positive genus (Guanhua Shao and Jiahua Zou, 2025)7Shao 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 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. 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, Homogeneous spaces · Same author Frederick Tsz-Ho Fong · Shared terms “self expanders”, “rotationally symmetric”, “principal curvatures”

  3. Uniqueness Theorems of Self-Conformal Solutions to Inverse Curvature Flows

    Nicholas Cheng-Hoong Chin, Frederick Tsz-Ho Fong and Jingbo WanWider flows

    Topics Inverse mean curvature flow, Homogeneous spaces, Other curvature flows · Same author Frederick Tsz-Ho Fong · Shared terms “homogeneous functions”, “round sphere”, “Euclidean space”

  4. 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, Boundaries & non-compact · Same author Frederick Tsz-Ho Fong · Shared terms “self expanders”, “rotationally symmetric”, “principal curvatures”

  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-shrinkers to the degree-one curvature flow with a rotationally symmetric conical end

    Siao-Hao GuoWider flows

    Topics Rotational symmetry & warped products, Homogeneous spaces, Self-shrinkers & translators · Shared terms “rotationally symmetric”, “principal curvatures”

  7. Self-expanders of positive genus

    Guanhua Shao and Jiahua ZouWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “self expanders”, “rotationally symmetric”

  8. 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

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. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 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.