Paper · Feb 11, 2017

Complete bounded -hypersurfaces in the weighted volume-preserving mean curvature flow

Yecheng Zhu, Yi Fang, Qing Chen

math.DGWider flowsarXiv:1702.03428
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Abstract

In this paper, we study the complete bounded -hypersurfaces in weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded -hypersurfaces with and get some applications of the volume comparison theorem. Secondly, we consider the relation among , extrinsic radius , intrinsic diameter , and dimension of the complete -hypersurface, and we obtain some estimates for the intrinsic diameter and the extrinsic radius. At last, we get some topological properties of the bounded -hypersurface with some natural and general restrictions.

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Submitted
Feb 11, 2017
Subjects
math.DG
Journal
Science China Mathematics-2018
DOI
10.1007/s11425-016-9020-9
Mean curvature flowVolume-preserving & constrained flows

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 Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type (Naoyuki Koike, 2016)6Koike 2016 Volume preserving mean curvature flow for star-shaped sets (Inwon Kim and Dohyun Kwon, 2018)7Kim 2018 Long time behaviour of discrete volume preserving mean curvature flows (Massimiliano Morini, Marcello Ponsiglione and Emanuele Spadaro, 2020)4Morini 2020 Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space (Zheng Huang, Longzhi Lin and Zhou Zhang, 2021)1Huang 2021 Weak-strong uniqueness for volume-preserving mean curvature flow (Tim Laux, 2022)5Laux 2022 Nonlocal estimates for the Volume Preserving Mean Curvature Flow and applications (Ben Lambert and Elena Mäder-Baumdicker, 2022)3Lambert 2022 Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus (Daniele De Gennaro, Antonia Diana, Andrea Kubin et al., 2023)8De Gennaro 2023 Asymptotic behavior of a diffused interface volume-preserving mean curvature flow (Matteo Bonforte, Francesco Maggi and Daniel Restrepo, 2024)2Bonforte 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space

    Zheng Huang, Longzhi Lin and Zhou ZhangWider flows

    Topics Volume-preserving & constrained flows

  2. Asymptotic behavior of a diffused interface volume-preserving mean curvature flow

    Matteo Bonforte, Francesco Maggi and Daniel RestrepoWider flows

    Topics Volume-preserving & constrained flows

  3. Nonlocal estimates for the Volume Preserving Mean Curvature Flow and applications

    Ben Lambert and Elena Mäder-BaumdickerWider flows

    Topics Volume-preserving & constrained flows

  4. Long time behaviour of discrete volume preserving mean curvature flows

    Massimiliano Morini, Marcello Ponsiglione and Emanuele SpadaroWider flows

    Topics Volume-preserving & constrained flows

  5. Weak-strong uniqueness for volume-preserving mean curvature flow

    Tim LauxWider flows

    Topics Volume-preserving & constrained flows

  6. Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type

    Naoyuki KoikeWider flows

    Topics Volume-preserving & constrained flows

  7. Volume preserving mean curvature flow for star-shaped sets

    Inwon Kim and Dohyun KwonWider flows

    Topics Volume-preserving & constrained flows

  8. Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus

    Daniele De Gennaro, Antonia Diana, Andrea Kubin et al.Wider flows

    Topics Volume-preserving & constrained flows

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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986)

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.