Paper · May 30, 2017

Convergence result and blow-up examples for the Guan–Li mean curvature flow on warped product spaces

Jérôme Vétois

math.APWider flowsarXiv:1705.10839v3
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Abstract

We examine the question of convergence of solutions to a geometric flow which was introduced by Guan and Li for starshaped hypersurfaces in space forms and generalized by Guan, Li, and Wang to the case of warped product spaces. We obtain a convergence result under a condition on the optimal modulus of continuity of the initial data. Moreover we show by examples that this condition is optimal at least in the one-dimensional case.

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Submitted
May 30, 2017
Last revised
Jun 18, 2019 (v3)
Subjects
math.AP
Journal
Communications in Analysis and Geometry 29 (2021), no. 8, 1917-1935
DOI
10.4310/CAG.2021.v29.n8.a6

Comments: Final version to appear in Communications in Analysis and Geometry

Mean curvature flowVolume-preserving & constrained flowsSingularitiesSphere theoremsRotational 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 Inverse mean curvature flows in warped product manifolds (Hengyu Zhou, 2016)3Zhou 2016 On the inverse mean curvature flow in warped product manifolds (Thomas Mullins, 2016)7Mullins 2016 Mean Curvature Type Flows of Graphs in Product Manifolds (Aijin Lin and Hengyu Zhou, 2017)4Lin 2017 Self-similar solutions of curvature flows in warped products (Shanze Gao and Hui Ma, 2018)5Gao 2018 Remarks on mean curvature flow solitons in warped products (Giulio Colombo, Luciano Mari and Marco Rigoli, 2018)6Colombo 2018 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)2Fong 2024 Isotropy invariant graphical mean curvature flows in warped products (Naotoshi Fujihara and Naoyuki Koike, 2024)8Fujihara 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 Sphere theorems, Rotational symmetry & warped products · Shared terms “space form”

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

    Frederick Tsz-Ho FongWider flows

    Topics Rotational symmetry & warped products · Shared terms “product spaces”

  3. Inverse mean curvature flows in warped product manifolds

    Hengyu ZhouWider flows

    Topics Rotational symmetry & warped products · Shared terms “starshaped”

  4. Mean Curvature Type Flows of Graphs in Product Manifolds

    Aijin Lin and Hengyu ZhouWider flows

    Topics Volume-preserving & constrained flows, Rotational symmetry & warped products

  5. Self-similar solutions of curvature flows in warped products

    Shanze Gao and Hui MaWider flows

    Topics Rotational symmetry & warped products

  6. Remarks on mean curvature flow solitons in warped products

    Giulio Colombo, Luciano Mari and Marco RigoliWider flows

    Topics Rotational symmetry & warped products

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

    Thomas MullinsWider flows

    Topics Rotational symmetry & warped products

  8. Isotropy invariant graphical mean curvature flows in warped products

    Naotoshi Fujihara and Naoyuki KoikeWider flows

    Topics Rotational symmetry & warped products, Singularities

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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  3. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.