Paper · Apr 28, 2016

Uniqueness of self-shrinkers to the degree-one curvature flow with a tangent cone at infinity

Siao-Hao Guo

math.DGWider flowsarXiv:1604.08577v2
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Abstract

Given a smooth, symmetric, homogeneous of degree one function satisfying for all , and an oriented, properly embedded smooth cone in , we show that under some suitable conditions on and the covariant derivatives of the second fundamental form of , there is at most one self-shrinker (i.e. an oriented hypersurface in for which holds, where is the position vector, is the unit normal vector, and are principal curvatures of ) that is asymptotic to the given cone at infinity.

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Submitted
Apr 28, 2016
Last revised
Nov 12, 2016 (v2)
Subjects
math.DG, math.AP
Other curvature flowsSelf-shrinkers & translatorsHomogeneous spacesStability & uniqueness

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 Existence of self-shrinkers to the degree-one curvature flow with a rotationally symmetric conical end (Siao-Hao Guo, 2016)1Guo 2016 Uniqueness of closed self-similar solutions to σₖ^α-curvature flow (Shanze Gao, Haizhong Li and Hui Ma, 2017)3Gao 2017 Asymptotic Behavior and Stability of Mean Curvature Flow with a Conical End (Siao-Hao Guo, 2019)7Guo 2019 Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders (Ilyas Khan, 2020)4Khan 2020 Rotationally symmetric translating solutions to extrinsic geometric flows (Sathyanarayanan Rengaswami, 2021)8Rengaswami 2021 Maximum Principles and Consequences for γ-translators in ℝⁿ⁺¹ (José Torres Santaella, 2023)2Santaella 2023 Uniqueness of blowups for forced mean curvature flow (Sven Hirsch and Jonathan J. Zhu, 2023)6Hirsch 2023 Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces (Frederick Tsz-Ho Fong, 2024)5Fong 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Existence of self-shrinkers to the degree-one curvature flow with a rotationally symmetric conical end

    Siao-Hao GuoWider flows

    Topics Homogeneous spaces, Self-shrinkers & translators, Other curvature flows · Same author Siao-Hao Guo · Shared terms “normal vector”, “position vector”, “unit normal”

  2. Maximum Principles and Consequences for -translators in

    José Torres SantaellaWider flows

    Topics Homogeneous spaces, Self-shrinkers & translators, Other curvature flows

  3. Uniqueness of closed self-similar solutions to -curvature flow

    Shanze Gao, Haizhong Li and Hui MaWider flows

    Topics Homogeneous spaces, Other curvature flows, Stability & uniqueness · Shared terms “principal curvatures”

  4. Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders

    Ilyas KhanWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “properly”

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

    Frederick Tsz-Ho FongWider flows

    Topics Homogeneous spaces, Self-shrinkers & translators, Other curvature flows · Shared terms “principal curvatures”

  6. Uniqueness of blowups for forced mean curvature flow

    Sven Hirsch and Jonathan J. ZhuWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “tangent cone”

  7. Asymptotic Behavior and Stability of Mean Curvature Flow with a Conical End

    Siao-Hao Guo

    Topics Self-shrinkers & translators, Stability & uniqueness · Same author Siao-Hao Guo

  8. Rotationally symmetric translating solutions to extrinsic geometric flows

    Sathyanarayanan RengaswamiWider flows

    Topics Homogeneous spaces, Self-shrinkers & translators, Other curvature flows · Shared terms “principal curvatures”

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.

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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) 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. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

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