Paper · Oct 17, 2016

On the inverse mean curvature flow in warped product manifolds

Thomas Mullins

math.DGWider flowsarXiv:1610.05234
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Abstract

We consider the warped product manifold, , with Riemannian metric , where is a smooth closed Riemannian -manifold. We investigate what sufficient curvature condition is required of to ensure that a solution to the inverse mean curvature flow - commencing with a star-shaped surface - exists for all times .

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Submitted
Oct 17, 2016
Subjects
math.DG

Comments: This result was obtained in the author's Master's thesis, submitted in December 2014 to the FU Berlin

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

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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 Inverse mean curvature flows in warped product manifolds (Hengyu Zhou, 2016)1Zhou 2016 Inverse mean curvature flow inside a cone in warped products (Li Chen, Jing Mao, Ni Xiang et al., 2017)2Chen 2017 Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds (Brian Allen, 2017)3Allen 2017 Mean Curvature Flows of Closed Hypersurfaces in Warped Product Manifolds (Zheng Huang, Zhou Zhang and Hengyu Zhou, 2017)5Huang 2017 Inverse curvature flows in Riemannian warped products (Julian Scheuer, 2017)4Scheuer 2017 Self similar solutions to the inverse mean curvature flow in ℝ² (Jui-En Chang, 2018)7Chang 2018 Inverse Mean Curvature Flow over Non-Star-Shaped Surfaces (Brian Harvie, 2019)8Harvie 2019 Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces (Frederick Tsz-Ho Fong, 2024)6Fong 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Inverse mean curvature flows in warped product manifolds

    Hengyu ZhouWider flows

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

  2. 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 · Shared terms “star shaped”

  3. 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

  4. Inverse curvature flows in Riemannian warped products

    Julian ScheuerWider flows

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

  5. 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

  6. Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces

    Frederick Tsz-Ho FongWider flows

    Topics Inverse mean curvature flow, Rotational symmetry & warped products, Other curvature flows · Shared terms “star shaped”

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

    Jui-En ChangWider flows

    Topics Inverse mean curvature flow, Other curvature flows

  8. Inverse Mean Curvature Flow over Non-Star-Shaped Surfaces

    Brian HarvieWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “shaped surface”, “star shaped”

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.