Paper · Feb 10, 2018

The blow-up of the conformal mean curvature flow

Xingxiao Li, Di Zhang

math.DGWider flowsarXiv:1802.03513
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Abstract

In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space . This kind of flow is a special case of a general modified mean curvature flow which is of various origination. As the main result, we prove a blow-up theorem concluding that, under the conformal mean curvature flow in , the maximum of the square norm of the second fundamental form of any compact submanifold tends to infinity in finite time. Furthermore, by using the idea of Andrews and Baker for studying the mean curvature flow of submanifolds in the Euclidean space, we also derive some more evolution formulas and inequalities which we believe to be useful in our further study of conformal mean curvature flow. Presently, these computations together with our main theorem are applied to provide a direct proof of a convergence theorem concluding that the external conformal forced mean curvature flow of a compact submanifold in with the same pinched condition as Andrews-Baker's will be convergent to a round point in finite time.

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Submitted
Feb 10, 2018
Subjects
math.DG

Comments: 47 pages including an appendix; all comments are welcome!

Mean curvature flowVolume-preserving & constrained flowsSingularities

Its neighbourhood

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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 Eternal forced mean curvature flows III - Morse homology (Graham Smith, 2016)5Smith 2016 Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type (Naoyuki Koike, 2016)4Koike 2016 Variational characterizations of ξ-submanifolds in the Eulicdean space bbr^(m+p) (Xingxiao Li and Zhaoping Li, 2016)8Li 2016 On the mean curvature flow of submanifolds in the standard Gaussian space ^† (An-Min Li, Xingxiao Li and Di Zhang, 2020)1Li 2020 Nonlocal estimates for the Volume Preserving Mean Curvature Flow and applications (Ben Lambert and Elena Mäder-Baumdicker, 2022)6Lambert 2022 A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting (Jiwoong Jang, 2023)7Jang 2023 Uniqueness of blowups for forced mean curvature flow (Sven Hirsch and Jonathan J. Zhu, 2023)2Hirsch 2023 Quenching for axisymmetric hypersurfaces under forced mean curvature flows (Hiroyoshi Mitake, Yusuke Oka and Hung Vinh Tran, 2024)3Mitake 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the mean curvature flow of submanifolds in the standard Gaussian space

    An-Min Li, Xingxiao Li and Di ZhangWider flows

    Topics Singularities · Same authors Xingxiao Li, Di Zhang · Shared terms “conformal mean”, “compact submanifold”, “square norm”

  2. Uniqueness of blowups for forced mean curvature flow

    Sven Hirsch and Jonathan J. ZhuWider flows

    Topics Volume-preserving & constrained flows, Singularities · Shared terms “forced mean”, “Euclidean space”

  3. Quenching for axisymmetric hypersurfaces under forced mean curvature flows

    Hiroyoshi Mitake, Yusuke Oka and Hung Vinh TranWider flows

    Topics Volume-preserving & constrained flows · Shared terms “forced mean”

  4. Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type

    Naoyuki KoikeWider flows

    Topics Volume-preserving & constrained flows, Singularities

  5. Eternal forced mean curvature flows III - Morse homology

    Graham SmithWider flows

    Topics Volume-preserving & constrained flows · Shared terms “forced mean”

  6. Nonlocal estimates for the Volume Preserving Mean Curvature Flow and applications

    Ben Lambert and Elena Mäder-BaumdickerWider flows

    Topics Volume-preserving & constrained flows, Singularities · Shared terms “finite time”, “Euclidean”

  7. A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting

    Jiwoong JangWider flows

    Topics Volume-preserving & constrained flows · Shared terms “forced mean”

  8. Variational characterizations of -submanifolds in the Eulicdean space

    Xingxiao Li and Zhaoping LiWider flows

    Topics Mean curvature flow · Same author Xingxiao Li · Shared terms “modified mean”, “Euclidean space”

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)

In the guidebook

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

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  2. 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.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.