Paper · Oct 4, 2016

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

Brian Allen

math.DGWider flowsarXiv:1610.01211v3
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Abstract

In this work we extend the ODE Maximum principle of Hamilton to non-compact hypersurfaces using the Omari-Yau maximum principle at infinity. As an application of this result, we investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over horospheres in and show long time existence of the flow as well as asymptotic convergence to horospheres.

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Submitted
Oct 4, 2016
Last revised
Apr 13, 2018 (v3)
Subjects
math.DG

Comments: 39 pages, substantial details added. arXiv admin note: text overlap with arXiv:1510.06670

Mean curvature flowOther curvature flowsInverse mean curvature flowBoundaries & 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 Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem and Riemannian Penrose Inequality Under L² Metric Convergence (Brian Allen, 2017)6Allen 2017 Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by Inverse Mean Curvature Flow (Brian Allen, 2017)3Allen 2017 Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds (Brian Allen, 2017)2Allen 2017 Inverse mean curvature evolution of entire graphs (Panagiota Daskalopoulos and Gerhard Huisken, 2017)8Daskalopoulo 2017 Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem (Brian Allen, 2018)7Allen 2018 Sobolev stability of the PMT and RPI using IMCF (Brian Allen, 2018)4Allen 2018 Evolution of non-compact hypersurfaces by inverse mean curvature (Beomjun Choi and Panagiota Daskalopoulos, 2018)1Choi 2018 Aleksandrov reflection for Geometric Flows in Hyperbolic Spaces (Theodora Bourni, José M. Espinar and Aakash Mishra, 2026)5Bourni 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Evolution of non-compact hypersurfaces by inverse mean curvature

    Beomjun Choi and Panagiota DaskalopoulosWider flows

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

  2. Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds

    Brian AllenWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Same author Brian Allen · Shared terms “IMCF”

  3. Stability of the Positive Mass Theorem and Riemannian Penrose Inequality for Asymptotically Hyperbolic Manifolds Foliated by Inverse Mean Curvature Flow

    Brian AllenWider flows

    Topics Inverse mean curvature flow, Boundaries & non-compact, Other curvature flows · Same author Brian Allen · Shared terms “IMCF”, “hyperbolic space”

  4. Sobolev stability of the PMT and RPI using IMCF

    Brian AllenWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Same author Brian Allen · Shared terms “IMCF”, “hyperbolic space”

  5. Aleksandrov reflection for Geometric Flows in Hyperbolic Spaces

    Theodora Bourni, José M. Espinar and Aakash MishraWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Boundaries & non-compact · Shared terms “hyperbolic space”, “horospheres”

  6. Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem and Riemannian Penrose Inequality Under Metric Convergence

    Brian AllenWider flows

    Topics Inverse mean curvature flow, Boundaries & non-compact, Other curvature flows · Same author Brian Allen · Shared terms “IMCF”

  7. Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem

    Brian AllenWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Same author Brian Allen · Shared terms “IMCF”

  8. 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

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.