Paper · Oct 29, 2021

Mean curvature type flow and sharp Micheal-Simon inequalities

J. Cui, P. Zhao

math.DGWider flowsarXiv:2111.00938
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Abstract

In this paper, we first investigate a new locally constrained mean curvature flow (1.5) and prove that if the initial hypersurface is of smoothly compact starshaped, then the solution of the flow (1.5) exists for all time and converges to a sphere in smooth topology. Following this flow argument, not only do we achieve a new proof of the celebrated sharp Michael-Simon inequality for mean curvature in (n+1) dimensional Euclidean space, but we also get the necessary and sufficient condition for the establishment of the equality. In the second part of this paper, we study a mean curvature type flow (1.7) of static convex hypersurfaces in (n+1) dimensional Euclidean space, and prove that the flow (1.7) has a unique smooth solution for all time t>0, and the static convexity of the hypersurface is preserved along the flow (1.7). Moreover, The solution of the flow (1.7) converges exponentially to a sphere of radius R in smooth topology as time tends to infinity. By exploiting the properties of this flow, we develop and present a new sharp Michael-Simon inequality for kth mean curvature.

Comments: 32 pages

Mean curvature flowVolume-preserving & constrained flowsLong-time existence & convergence

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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 2021, 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 Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space (Zheng Huang, Longzhi Lin and Zhou Zhang, 2021)4Huang 2021 Locally constrained flows and sharp Michael-Simon inequalities in hyperbolic space (Jingshi Cui and Peibiao Zhao, 2022)1Cui 2022 Michael-Simon type inequalities in hyperbolic space ℍⁿ⁺¹ via Brendle-Guan-Li's flows (Jingshi Cui and Peibiao Zhao, 2022)2Cui 2022 Horizontal inverse mean curvature flow in the Heisenberg group (Jingshi Cui and Peibiao Zhao, 2023)7Cui 2023 General Conformally Induced Mean Curvature Flow (Joshua Flynn and Jacob Reznikov, 2023)8Flynn 2023 The weighted geometric inequalities for static convex domains in static rotationally symmetric spaces (Shujing Pan and Bo Yang, 2023)5Pan 2023 Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds (Jingshi Cui and Peibiao Zhao, 2025)3Cui 2025 Existence of the classical solution to the fractional mean curvature flow with capillary-type boundary conditions (Linlin Fan and Peibiao Zhao, 2026)6Fan 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Locally constrained flows and sharp Michael-Simon inequalities in hyperbolic space

    Jingshi Cui and Peibiao ZhaoWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Same authors J. Cui, P. Zhao · Shared terms “Simon inequality”, “Michael Simon”, “sharp Michael”

  2. Michael-Simon type inequalities in hyperbolic space via Brendle-Guan-Li's flows

    Jingshi Cui and Peibiao ZhaoWider flows PDF

    Topics Volume-preserving & constrained flows · Same authors J. Cui, P. Zhao · Shared terms “Michael Simon”, “Simon inequality”, “sharp Michael”

  3. Legendrian curve flow in Sasakian sub-Riemannian 3-manifolds

    Jingshi Cui and Peibiao ZhaoWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Same authors J. Cui, P. Zhao · Shared terms “exploiting”, “Euclidean”

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

    Zheng Huang, Longzhi Lin and Zhou ZhangWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “initial hypersurface”, “Euclidean space”, “converges exponentially”

  5. The weighted geometric inequalities for static convex domains in static rotationally symmetric spaces

    Shujing Pan and Bo YangWider flows PDF

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “static convex”, “smooth topology”, “initial hypersurface”

  6. Existence of the classical solution to the fractional mean curvature flow with capillary-type boundary conditions

    Linlin Fan and Peibiao ZhaoWider flows PDF

    Topics Volume-preserving & constrained flows · Same author P. Zhao · Shared terms “type flow”, “curvature type”

  7. Horizontal inverse mean curvature flow in the Heisenberg group

    Jingshi Cui and Peibiao ZhaoWider flows PDF

    Topics Mean curvature flow · Same authors J. Cui, P. Zhao · Shared terms “following”

  8. General Conformally Induced Mean Curvature Flow

    Joshua Flynn and Jacob ReznikovWider flows PDF

    Topics Volume-preserving & constrained flows · Shared terms “type flow”, “curvature type”, “starshaped”

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. 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)

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.