Paper · Oct 5, 2017

The mean curvature flow by parallel hypersurfaces

Hiuri Fellipe Santos dos Reis, Keti Tenenblat

math.DGWider flowsarXiv:1710.02122
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Abstract

It is shown that a hypersurface of a space form is the initial data for a solution to the mean curvature flow by parallel hypersurfaces if, and only if, it is isoparametric. By solving an ordinary differential equation, explicit solutions are given for all isoparametric hypersurfaces of space forms. In particular, for such hypersurfaces of the sphere, the exact collapsing time into a focal submanifold is given in terms of its dimension, the principal curvatures and their multiplicities.

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Submitted
Oct 5, 2017
Subjects
math.DG
Mean curvature flow(Non)collapsingSphere theorems

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the inverse mean curvature flow by parallel hypersurfaces in space forms

    Alancoc dos Santos Alencar and Keti TenenblatWider flows

    Topics Sphere theorems, (Non)collapsing · Same author Keti Tenenblat · Shared terms “parallel hypersurfaces”, “space form”, “isoparametric”

  2. Isoparametric hypersurfaces of Riemannian manifolds as initial data for the mean curvature flow

    Felippe Guimarães, João Batista Marques dos Santos and João Paulo dos SantosWider flows

    Topics Sphere theorems · Shared terms “isoparametric hypersurfaces”, “space form”, “initial data”

  3. Collapsing of Mean Curvature Flow of Hypersurfaces to Complex Submanifolds

    Farnaz Ghanbari and SamreenaWider flows

    Topics (Non)collapsing

  4. Pyramid Ricci Flow in Higher Dimensions

    Andrew D. McLeod and Peter M. Topping

    Topics Sphere theorems, (Non)collapsing

  5. Surfaces pinched by normal curvature for mean curvature flow in space forms

    Dong Pu, Jingjing Su and Hongwei XuWider flows

    Topics Sphere theorems · Shared terms “space form”

  6. Real hypersurfaces with -Ricci solitons of non-flat complex space forms

    Xiaomin Chen

    Topics Sphere theorems · Shared terms “space form”, “principal curvatures”

  7. Graphical translating solitons for the mean curvature flow and isoparametric functions

    Tomoki FujiiWider flows

    Topics Mean curvature flow · Shared terms “isoparametric”, “ordinary differential”, “differential equation”

  8. A sharp convergence theorem for the mean curvature flow in spheres I

    Dong PuWider flows

    Topics Sphere theorems

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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) 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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  3. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.