Paper · Oct 26, 2016

Contracting convex hypersurfaces by functions of the mean curvature

Shunzi Guo

math.DGWider flowsarXiv:1610.08209
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Abstract

This paper concerns the evolution of a closed convex hypersurface in , in direction of its inner unit normal vector, where the speed is given by a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow exists on a finite maximal interval, convexity is preserved and the hypersurfaces shrink down to a single point as the final time is approached. This result covers and generalises the corresponding result of Schulze for the positive power mean curvature flow to a much larger possible class of flows by the functions depending only on the mean curvature.

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Submitted
Oct 26, 2016
Subjects
math.DG

Comments: 19pages, Comments are welcome

Mean curvature flow

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 On the classification of ancient solutions to curvature flows on the sphere (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2016)7Bryan 2016 Volume preserving non homogeneous mean curvature flow of convex hypersurfaces (Maria Chiara Bertini and Carlo Sinestrari, 2016)5Bertini 2016 Mixed volume preserving flow by powers of homogeneous curvature functions of degree one (Shunzi Guo, 2016)1Guo 2016 Contraction of surfaces in hyperbolic space and in sphere (Yingxiang Hu, Haizhong Li, Yong Wei et al., 2019)3Hu 2019 Contracting convex surfaces by mean curvature flow with free boundary on convex barriers (Sven Hirsch and Martin Li, 2020)2Hirsch 2020 A Pinching Estimate for Convex Hypersurfaces Evolving Under a Nonhomogeneous Variant of Mean Curvature Flow (Tim Espin, 2020)8Espin 2020 Contracting self-similar solutions of nonhomogeneous curvature flows (James McCoy, 2020)6McCoy 2020 Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow (Carlo Sinestrari and Liangjun Weng, 2024)4Sinestrari 2024
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

    Same author Shunzi Guo · Shared terms “convex hypersurfaces”, “generalises”

  2. Contracting convex surfaces by mean curvature flow with free boundary on convex barriers

    Sven Hirsch and Martin LiWider flows

    Topics Mean curvature flow · Shared terms “contracting convex”, “convex hypersurfaces”, “depending”

  3. Contraction of surfaces in hyperbolic space and in sphere

    Yingxiang Hu, Haizhong Li, Yong Wei et al.Wider flows

    Shared terms “positive power”, “final time”, “contracting”

  4. Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow

    Carlo Sinestrari and Liangjun WengWider flows

    Topics Mean curvature flow · Shared terms “power mean”, “positive power”, “flow exists”

  5. Volume preserving non homogeneous mean curvature flow of convex hypersurfaces

    Maria Chiara Bertini and Carlo SinestrariWider flows

    Topics Mean curvature flow · Shared terms “convex hypersurfaces”, “closed convex”

  6. Contracting self-similar solutions of nonhomogeneous curvature flows

    James McCoyWider flows

    Shared terms “contracting”, “convex hypersurfaces”

  7. On the classification of ancient solutions to curvature flows on the sphere

    Paul Bryan, Mohammad N. Ivaki and Julian ScheuerWider flows

    Topics Mean curvature flow · Shared terms “smooth function”, “homogeneity”, “restriction”

  8. A Pinching Estimate for Convex Hypersurfaces Evolving Under a Nonhomogeneous Variant of Mean Curvature Flow

    Tim EspinWider flows

    Topics Mean curvature flow · Shared terms “convex hypersurfaces”, “preserved”

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)

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.