Paper · Jul 11, 2016

Collapse of the mean curvature flow for certain kind of invariant hypersurfaces in a Hilbert space

Naoyuki Koike

math.DGWider flowsarXiv:1607.02899v4
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Abstract

In this paper, we investigate the regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are regularized minimal. We prove that, if the invariant hypersurface satisfies a certain kind of horizontally convexity condition, then it collapses to an orbit of the Hilbert Lie group action along the regularized mean curvature flow.

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Submitted
Jul 11, 2016
Last revised
Nov 6, 2018 (v4)
Subjects
math.DG

Comments: I found some errors in this paper. Hence I withdrawal this paper

Mean curvature flowHomogeneous spaces

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  4. Cohomogeneity one Hermann actions and backward mean curvature flows

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  6. Mean curvature flow for pinched submanifolds in rank one symmetric spaces

    Naoyuki Koike, Yoshiyuki Mizumura and Nana UenoyamaWider flows

    Topics Mean curvature flow · Same author Naoyuki Koike · Shared terms “collapse”, “flow starting”

  7. Isotropy invariant graphical mean curvature flows in warped products

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

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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) 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)

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.