Paper · Feb 27, 2018

Parabolicity, Brownian escape rate and properness of self-similar solutions of the direct and inverse Mean Curvature Flow

Vicent Gimeno, Vicente Palmer

math.DGWider flowsarXiv:1802.09833
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Abstract

We study some potential theoretic properties of homothetic solitons of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in , we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that parabolic MCF-solitons with are self-shrinkers and that parabolic IMCF-solitons of any dimension are self-expanders. We have studied too the geometric behavior of parabolic MCF and IMCF-solitons confined in a ball, the behavior of the Mean Exit Time function for the Brownian motion defined on as well as a classification of properly immersed MCF-self-shrinkers with bounded second fundamental form, following the lines of.

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Submitted
Feb 27, 2018
Subjects
math.DG

Comments: 32 pages

Mean curvature flowOther curvature flowsInverse mean curvature flowSelf-shrinkers & translators

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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. Self similar solutions to the inverse mean curvature flow in

    Jui-En ChangWider flows

    Topics Inverse mean curvature flow, Other curvature flows

  2. Rotational symmetry of self-expanders to the inverse mean curvature flow with cylindrical ends

    Gregory Drugan, Frederick Tsz-Ho Fong and Hojoo LeeWider flows

    Topics Inverse mean curvature flow, Self-shrinkers & translators, Other curvature flows · Shared terms “self expanders”

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

    Tomoki FujiiWider flows

    Topics Inverse mean curvature flow, Self-shrinkers & translators, Other curvature flows

  4. Another proof of the existence of homothetic solitons of the inverse mean curvature flow

    Shu-Yu HsuWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “homothetic solitons”

  5. Self-Expanders to Inverse Curvature Flows by Homogeneous Functions

    Tsz-Kiu Aaron Chow, Ka-Wing Chow and Frederick Tsz-Ho FongWider flows

    Topics Inverse mean curvature flow, Self-shrinkers & translators, Other curvature flows · Shared terms “self expanders”

  6. Isoperimetry and the properness of weak inverse mean curvature flow

    Kai XuWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “properness”

  7. Existence of self-similar solution of the inverse mean curvature flow

    K. M. HuiWider flows

    Topics Inverse mean curvature flow, Other curvature flows

  8. Classification of ruled surfaces as homothetic self-similar solutions of the inverse mean curvature flow in the Lorentz-Minkowski 3-space

    Gregório Silva Neto and Vanessa SilvaWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “homothetic solitons”

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) 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) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990)

In the guidebook

  1. 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.