Paper · Mar 21, 2017

Expansion of pinched hypersurfaces of the Euclidean and hyperbolic space by high powers of curvature

Heiko Kröner, Julian Scheuer

math.DGWider flowsarXiv:1703.07087v3
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Abstract

We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form , where and is a positive, strictly monotone and 1-homogeneous curvature function. In particular this class includes the mean curvature . We prove that a certain initial pinching condition is preserved and the properly rescaled hypersurfaces converge smoothly to the unit sphere. We show that an example due to Andrews-McCoy-Zheng can be used to construct strictly convex initial hypersurfaces, for which the inverse mean curvature flow to the power loses convexity, justifying the necessity to impose a certain pinching condition on the initial hypersurface.

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Submitted
Mar 21, 2017
Last revised
Aug 3, 2018 (v3)
Subjects
math.DG, math.AP
Journal
Math. Nachr. 292, no. 7, p. 1514-1529, (2019)
DOI
10.1002/mana.201700370

Comments: 18 pages. We included an example for the loss of convexity and pinching. In the third version we dropped the concavity assumption on F. Comments are welcome

Mean curvature flowOther curvature flowsInverse mean curvature flowCurvature pinchingHomogeneous spacesLong-time existence & convergence

Its neighbourhood

Connected papers

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 Mixed volume preserving flow by powers of homogeneous curvature functions of degree one (Shunzi Guo, 2016)1Guo 2016 Self-similar solutions of σₖ^α-curvature flow (Shanze Gao and Hui Ma, 2016)8Gao 2016 New pinching estimates for Inverse curvature flows in space forms (Yong Wei, 2017)7Wei 2017 Inverse curvature flows in Riemannian warped products (Julian Scheuer, 2017)3Scheuer 2017 Li-Yau gradient estimates for curvature flows in positively curved manifolds (Paul Bryan, Heiko Kröner and Julian Scheuer, 2019)2Bryan 2019 Contracting axially symmetric hypersurfaces by powers of the σₖ-curvature (Haizhong Li, Xianfeng Wang and Jing Wu, 2019)4Li 2019 Shifted inverse curvature flows in hyperbolic space (Xianfeng Wang, Yong Wei and Tailong Zhou, 2020)6Wang 2020 Self-similar solutions to fully nonlinear curvature flows by high powers of curvature (Shanze Gao, Haizhong Li and Xianfeng Wang, 2020)5Gao 2020
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mixed volume preserving flow by powers of homogeneous curvature functions of degree one

    Shunzi GuoWider flows

    Topics Curvature pinching, Homogeneous spaces, Long-time existence & convergence · Shared terms “pinching condition”, “curvature function”, “initial hypersurface”

  2. Li-Yau gradient estimates for curvature flows in positively curved manifolds

    Paul Bryan, Heiko Kröner and Julian ScheuerWider flows

    Topics Other curvature flows · Same authors Heiko Kröner, Julian Scheuer · Shared terms “initial hypersurface”, “strictly convex”

  3. Inverse curvature flows in Riemannian warped products

    Julian ScheuerWider flows

    Topics Inverse mean curvature flow, Long-time existence & convergence, Other curvature flows · Same author Julian Scheuer · Shared terms “expanding curvature”, “curvature function”

  4. Contracting axially symmetric hypersurfaces by powers of the -curvature

    Haizhong Li, Xianfeng Wang and Jing WuWider flows

    Topics Curvature pinching, Long-time existence & convergence, Other curvature flows · Shared terms “rescaled hypersurfaces”, “properly rescaled”, “pinching condition”

  5. Self-similar solutions to fully nonlinear curvature flows by high powers of curvature

    Shanze Gao, Haizhong Li and Xianfeng WangWider flows

    Topics Homogeneous spaces, Other curvature flows · Shared terms “high powers”, “strictly convex”

  6. Shifted inverse curvature flows in hyperbolic space

    Xianfeng Wang, Yong Wei and Tailong ZhouWider flows

    Topics Inverse mean curvature flow, Homogeneous spaces, Other curvature flows · Shared terms “hyperbolic space”, “curvature function”

  7. New pinching estimates for Inverse curvature flows in space forms

    Yong WeiWider flows

    Topics Curvature pinching, Inverse mean curvature flow, Homogeneous spaces · Shared terms “strictly convex”

  8. Self-similar solutions of -curvature flow

    Shanze Gao and Hui MaWider flows

    Topics Curvature pinching, Homogeneous spaces, Other curvature flows · Shared terms “pinching condition”, “strictly convex”

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) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 6A.9
    Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives

    Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.

  2. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.

  3. 11A.10
    Hamilton–Ivey PinchingBook 11A · The Ricci Flow: Existence and Maximum Principles

    Why high curvature in dimension 3 is almost positive.