Paper · Nov 12, 2018

Evolution of non-compact hypersurfaces by inverse mean curvature

Beomjun Choi, Panagiota Daskalopoulos

math.DGWider flowsarXiv:1811.04594v3
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Abstract

We study the evolution of complete non-compact convex hypersurfaces in by the inverse mean curvature flow. We establish the long time existence of solutions and provide the characterization of the maximal time of existence in terms of the tangent cone at infinity of the initial hypersurface. Our proof is based on an a'priori pointwise estimate on the mean curvature of the solution from below in terms of the aperture of a supporting cone at infinity. The strict convexity of convex solutions is shown by means of viscosity solutions. Our methods also give an alternative proof of the result by Huisken and Ilmanen on compact star-shaped solutions, based on maximum principle argument.

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Submitted
Nov 12, 2018
Last revised
Oct 22, 2020 (v3)
Subjects
math.DG

Comments: 36 pages, 5 figures. In the second version, Section 4 has been rewritten with more details and Appendix A.3 has been added. There are some minor changes like correcting errata

Mean curvature flowOther curvature flowsInverse mean curvature flowWeak & singular flowsBoundaries & non-compactLong-time existence & convergence

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 ODE Maximum Principle at Infinity and Non-Compact Solutions of IMCF in Hyperbolic Space (Brian Allen, 2016)3Allen 2016 Inverse mean curvature evolution of entire graphs (Panagiota Daskalopoulos and Gerhard Huisken, 2017)2Daskalopoulo 2017 Self similar solutions to the inverse mean curvature flow in ℝ² (Jui-En Chang, 2018)6Chang 2018 Inverse Mean Curvature Flow with Singularities (Beomjun Choi and Pei-Ken Hung, 2018)1Choi 2018 Convergence of Gauss curvature flows to translating solitons (Beomjun Choi, Kyeongsu Choi and Panagiota Daskalopoulos, 2019)4Choi 2019 Inverse Mean Curvature Flow over Non-Star-Shaped Surfaces (Brian Harvie, 2019)7Harvie 2019 A class of inverse curvature flows for star-shaped hypersurfaces evolving in a cone (Jing Mao and Qiang Tu, 2021)5Mao 2021 On the evolution of hypersurfaces along their inverse spacetime mean curvature (Gerhard Huisken and Markus Wolff, 2022)8Huisken 2022
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Inverse Mean Curvature Flow with Singularities

    Beomjun Choi and Pei-Ken HungWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Same author Beomjun Choi · Shared terms “tangent cone”, “convex hypersurfaces”

  2. Inverse mean curvature evolution of entire graphs

    Panagiota Daskalopoulos and Gerhard HuiskenWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Boundaries & non-compact · Same author Panagiota Daskalopoulos · Shared terms “maximal time”

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

    Brian AllenWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Boundaries & non-compact · Shared terms “compact hypersurfaces”, “maximum principle”

  4. Convergence of Gauss curvature flows to translating solitons

    Beomjun Choi, Kyeongsu Choi and Panagiota DaskalopoulosWider flows

    Topics Boundaries & non-compact, Other curvature flows · Same authors Beomjun Choi, Panagiota Daskalopoulos · Shared terms “compact convex”, “initial hypersurface”, “convex hypersurfaces”

  5. A class of inverse curvature flows for star-shaped hypersurfaces evolving in a cone

    Jing Mao and Qiang TuWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Shared terms “star shaped”, “convex hypersurfaces”

  6. Self similar solutions to the inverse mean curvature flow in

    Jui-En ChangWider flows

    Topics Inverse mean curvature flow, Other curvature flows

  7. Inverse Mean Curvature Flow over Non-Star-Shaped Surfaces

    Brian HarvieWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Other curvature flows · Shared terms “star shaped”

  8. On the evolution of hypersurfaces along their inverse spacetime mean curvature

    Gerhard Huisken and Markus WolffWider flows

    Topics Inverse mean curvature flow, Weak & singular flows, Boundaries & non-compact · Shared terms “Ilmanen”, “Huisken”

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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  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.