Paper · Sep 30, 2016

Inverse mean curvature flows in warped product manifolds

Hengyu Zhou

math.DGWider flowsarXiv:1609.09665v5
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Abstract

We study inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds with a positive warping factor . If and , we show that these flows exist for all times, remain starshaped and mean convex. Plus the positivity of and a curvature condition we obtain a lower positive bound of mean curvature along these flows independent of the initial mean curvature. We also give a sufficient condition to extend the asymptotic behavior of these flows in Euclidean spaces into some more general warped product manifolds.

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Submitted
Sep 30, 2016
Last revised
Aug 5, 2017 (v5)
Subjects
math.DG
DOI
10.1007/s12220-017-9887-z

Comments: Version 4: Final Version. Update some remarks for the conditions(see Remark 5.4). To appear the Journal of Geometric Analysis. Version 5: correct a reference typo and add two new references, mention an unpublished work in this direction

Mean curvature flowOther curvature flowsInverse mean curvature flowLong-time existence & convergenceRotational symmetry & warped products

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 inverse mean curvature flow in warped product manifolds (Thomas Mullins, 2016)1Mullins 2016 Inverse mean curvature flow inside a cone in warped products (Li Chen, Jing Mao, Ni Xiang et al., 2017)4Chen 2017 Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds (Brian Allen, 2017)5Allen 2017 Mean Curvature Flows of Closed Hypersurfaces in Warped Product Manifolds (Zheng Huang, Zhou Zhang and Hengyu Zhou, 2017)2Huang 2017 Mean Curvature Type Flows of Graphs in Product Manifolds (Aijin Lin and Hengyu Zhou, 2017)3Lin 2017 Inverse curvature flows in Riemannian warped products (Julian Scheuer, 2017)6Scheuer 2017 Self similar solutions to the inverse mean curvature flow in ℝ² (Jui-En Chang, 2018)8Chang 2018 Mean curvature flows of graphs sliding off to infinity in warped product manifolds (Naotoshi Fujihara, 2023)7Fujihara 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the inverse mean curvature flow in warped product manifolds

    Thomas MullinsWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Long-time existence & convergence · Shared terms “curvature condition”, “sufficient”

  2. Mean Curvature Flows of Closed Hypersurfaces in Warped Product Manifolds

    Zheng Huang, Zhou Zhang and Hengyu ZhouWider flows

    Topics Rotational symmetry & warped products, Long-time existence & convergence · Same author Hengyu Zhou

  3. Mean Curvature Type Flows of Graphs in Product Manifolds

    Aijin Lin and Hengyu ZhouWider flows

    Topics Rotational symmetry & warped products, Long-time existence & convergence · Same author Hengyu Zhou

  4. Inverse mean curvature flow inside a cone in warped products

    Li Chen, Jing Mao, Ni Xiang et al.Wider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Long-time existence & convergence

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

    Brian AllenWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Long-time existence & convergence

  6. Inverse curvature flows in Riemannian warped products

    Julian ScheuerWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Long-time existence & convergence

  7. Mean curvature flows of graphs sliding off to infinity in warped product manifolds

    Naotoshi FujiharaWider flows

    Topics Rotational symmetry & warped products, Long-time existence & convergence

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

    Jui-En ChangWider flows

    Topics Inverse mean curvature flow, Other curvature 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) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.