Paper · Sep 19, 2017

Inverse mean curvature evolution of entire graphs

Panagiota Daskalopoulos, Gerhard Huisken

math.DGWider flowsarXiv:1709.06665
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Abstract

We study the evolution of strictly mean-convex entire graphs over by Inverse Mean Curvature flow. First we establish the global existence of starshaped entire graphs with superlinear growth at infinity. The main result in this work concerns the critical case of asymptotically conical entire convex graphs. In this case we show that there exists a time , which depends on the growth at infinity of the initial data, such that the unique solution of the flow exists for all . Moreover, as the solution converges to a flat plane. Our techniques exploit the ultra-fast diffusion character of the fully-nonlinear flow, a property that implies that the asymptotic behavior at spatial infinity of our solution plays a crucial influence on the maximal time of existence, as such behavior propagates infinitely fast towards the interior.

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Submitted
Sep 19, 2017
Subjects
math.DG

Comments: 42 pages, 2 figures

Mean curvature flowOther curvature flowsInverse mean curvature flowBoundaries & non-compactLong-time existence & convergence

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  3. Inverse mean curvature flow and Ricci-pinched three-manifolds

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  5. Inverse mean curvature flows in warped product manifolds

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  6. ODE Maximum Principle at Infinity and Non-Compact Solutions of IMCF in Hyperbolic Space

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    Topics Inverse mean curvature flow, Long-time existence & convergence, Boundaries & non-compact

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

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  8. Uniqueness of entire graphs evolving by Mean Curvature flow

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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) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.