Paper · Dec 19, 2017

Mean Curvature Type Flows of Graphs in Product Manifolds

Aijin Lin, Hengyu Zhou

math.DGWider flowsarXiv:1712.06838
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Abstract

In this note we study a large class of mean curvature type flows of graphs in product manifold where N is a closed Riemann- ian manifold. Their speeds are the mean curvature of graphs plus a prescribed function. We establish long time existence and uniformly convergence of those flows with a barrier condition and a condition on the derivative of prescribed function with respect to the height. As an application we construct a weighted mean curvature flow in large classes of warped product manifolds which evolves each graph into a totally ge- odesic slice

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Submitted
Dec 19, 2017
Subjects
math.DG, math.AP

Comments: Page 14. This work is a noboundary case of the paper arXiv:1702.02449

Mean curvature flowVolume-preserving & constrained flowsLong-time existence & convergenceRotational symmetry & warped products

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 Inverse mean curvature flows in warped product manifolds (Hengyu Zhou, 2016)3Zhou 2016 On the inverse mean curvature flow in warped product manifolds (Thomas Mullins, 2016)6Mullins 2016 Nonparametric mean curvature type flows of graphs with contact angle conditions (Hengyu Zhou, 2017)1Zhou 2017 Convergence result and blow-up examples for the Guan–Li mean curvature flow on warped product spaces (Jérôme Vétois, 2017)8Vétois 2017 Long Time Existence of IMCF on Metrics Conformal to Warped Product Manifolds (Brian Allen, 2017)7Allen 2017 Mean Curvature Flows of Closed Hypersurfaces in Warped Product Manifolds (Zheng Huang, Zhou Zhang and Hengyu Zhou, 2017)2Huang 2017 Mean curvature type flow and sharp Micheal-Simon inequalities (J. Cui and P. Zhao, 2021)4Cui 2021 Mean curvature flows of graphs sliding off to infinity in warped product manifolds (Naotoshi Fujihara, 2023)5Fujihara 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Nonparametric mean curvature type flows of graphs with contact angle conditions

    Hengyu ZhouWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Same author Hengyu Zhou · Shared terms “type flow”, “curvature type”, “uniformly convergence”

  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. Inverse mean curvature flows in warped product manifolds

    Hengyu ZhouWider flows

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

  4. Mean curvature type flow and sharp Micheal-Simon inequalities

    J. Cui and P. ZhaoWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “type flow”, “curvature type”

  5. 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

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

    Thomas MullinsWider flows

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

  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. Convergence result and blow-up examples for the Guan–Li mean curvature flow on warped product spaces

    Jérôme VétoisWider flows

    Topics Volume-preserving & constrained flows, 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) 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) 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)

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.