Paper · Aug 21, 2017

Mean Curvature Flows of Closed Hypersurfaces in Warped Product Manifolds

Zheng Huang, Zhou Zhang, Hengyu Zhou

math.DGWider flowsarXiv:1708.06049v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We investigate the mean curvature flows in a class of warped product manifolds with closed hypersurfaces fibering over . In particular, we prove that under natural conditions on the warping function and Ricci curvature bound for the ambient space, there exists a large class of closed initial hypersurfaces, as geodesic graphs over the totally geodesic hypersurface , such that the mean curvature flow starting from exists for all time and converges to .

View PDF View HTML arXiv page
Submitted
Aug 21, 2017
Last revised
May 17, 2019 (v2)
Subjects
math.DG, math.AP

Comments: final version, to appear Math. Research Letters

Mean curvature flowLong-time existence & convergenceRotational symmetry & warped productsDiscrete & graph Ricci

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 Mean curvature flow in Fuchsian manifolds (Zheng Huang, Longzhi Lin and Zhou Zhang, 2016)1Huang 2016 Inverse mean curvature flows in warped product manifolds (Hengyu Zhou, 2016)2Zhou 2016 On the inverse mean curvature flow in warped product manifolds (Thomas Mullins, 2016)5Mullins 2016 Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds (Brian Allen, 2017)7Allen 2017 Mean Curvature Type Flows of Graphs in Product Manifolds (Aijin Lin and Hengyu Zhou, 2017)4Lin 2017 Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space (Zheng Huang, Longzhi Lin and Zhou Zhang, 2021)6Huang 2021 Closed self-similar solutions to flows by negative powers of curvature (Shanze Gao, 2022)8Gao 2022 Mean curvature flows of graphs sliding off to infinity in warped product manifolds (Naotoshi Fujihara, 2023)3Fujihara 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean curvature flow in Fuchsian manifolds

    Zheng Huang, Longzhi Lin and Zhou ZhangWider flows

    Topics Rotational symmetry & warped products, Long-time existence & convergence · Same authors Zheng Huang, Zhou Zhang · Shared terms “closed initial”, “geodesic graph”, “totally geodesic”

  2. Inverse mean curvature flows in warped product manifolds

    Hengyu ZhouWider flows

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

  3. 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 · Shared terms “geodesic graph”, “warping function”

  4. 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

  5. On the inverse mean curvature flow in warped product manifolds

    Thomas MullinsWider flows

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

  6. Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space

    Zheng Huang, Longzhi Lin and Zhou ZhangWider flows

    Topics Long-time existence & convergence · Same authors Zheng Huang, Zhou Zhang · Shared terms “initial hypersurface”

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

    Brian AllenWider flows

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

  8. Closed self-similar solutions to flows by negative powers of curvature

    Shanze GaoWider flows

    Topics Rotational symmetry & warped products

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) 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994) Discrete & graph Ricci Curvature and Ricci flow for graphs, networks and triangulated surfaces. It runs from Chow–Luo's combinatorial flow on circle packings to Ollivier's… Start with: Combinatorial Ricci Flows on Surfaces (Chow et al., 2002)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 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.