Paper · Feb 10, 2018

Self-similar solutions of curvature flows in warped products

Shanze Gao, Hui Ma

math.DGWider flowsarXiv:1802.03521
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Abstract

In this paper we study self-similar solutions in warped products satisfying , where is a nonnegative constant and is in a class of general curvature functions including powers of mean curvature and Gauss curvature. We show that slices are the only closed strictly convex self-similar solutions in the hemisphere for such . We also obtain a similar uniqueness result in hyperbolic space for Gauss curvature and .

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Submitted
Feb 10, 2018
Subjects
math.DG, math.AP

Comments: 19 pages

Other curvature flowsStability & uniquenessRotational symmetry & warped products

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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 On the inverse mean curvature flow in warped product manifolds (Thomas Mullins, 2016)7Mullins 2016 Self-similar solutions of σₖ^α-curvature flow (Shanze Gao and Hui Ma, 2016)3Gao 2016 Uniqueness of closed self-similar solutions to σₖ^α-curvature flow (Shanze Gao, Haizhong Li and Hui Ma, 2017)2Gao 2017 Inverse curvature flows in Riemannian warped products (Julian Scheuer, 2017)8Scheuer 2017 Remarks on mean curvature flow solitons in warped products (Giulio Colombo, Luciano Mari and Marco Rigoli, 2018)5Colombo 2018 Self-similar solutions to fully nonlinear curvature flows by high powers of curvature (Shanze Gao, Haizhong Li and Xianfeng Wang, 2020)4Gao 2020 Closed self-similar solutions to flows by negative powers of curvature (Shanze Gao, 2022)1Gao 2022 Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces (Frederick Tsz-Ho Fong, 2024)6Fong 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Closed self-similar solutions to flows by negative powers of curvature

    Shanze GaoWider flows

    Topics Rotational symmetry & warped products, Other curvature flows · Same author Shanze Gao · Shared terms “Gauss curvature”, “curvature function”, “strictly convex”

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

    Shanze Gao, Haizhong Li and Hui MaWider flows

    Topics Other curvature flows, Stability & uniqueness · Same authors Shanze Gao, Hui Ma · Shared terms “strictly convex”

  3. Self-similar solutions of -curvature flow

    Shanze Gao and Hui MaWider flows

    Topics Other curvature flows · Same authors Shanze Gao, Hui Ma · Shared terms “closed strictly”, “strictly convex”

  4. Self-similar solutions to fully nonlinear curvature flows by high powers of curvature

    Shanze Gao, Haizhong Li and Xianfeng WangWider flows

    Topics Other curvature flows · Same author Shanze Gao · Shared terms “closed strictly”, “convex self”, “strictly convex”

  5. Remarks on mean curvature flow solitons in warped products

    Giulio Colombo, Luciano Mari and Marco RigoliWider flows

    Topics Rotational symmetry & warped products, Stability & uniqueness · Shared terms “Gauss”

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

    Frederick Tsz-Ho FongWider flows

    Topics Rotational symmetry & warped products, Other curvature flows, Stability & uniqueness

  7. On the inverse mean curvature flow in warped product manifolds

    Thomas MullinsWider flows

    Topics Rotational symmetry & warped products, Other curvature flows

  8. Inverse curvature flows in Riemannian warped products

    Julian ScheuerWider flows

    Topics Rotational symmetry & warped products, Other curvature flows · Shared terms “curvature function”

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) 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)