Paper · Sep 26, 2023

General Conformally Induced Mean Curvature Flow

Joshua Flynn, Jacob Reznikov

math.DGWider flowsarXiv:2309.14679
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Abstract

This paper continues the investigation of isoperimetric inequalities through volume preserving and area decreasing mean curvature type flows related to conformal Killing vector fields. Results of this kind prior to this paper all studied convex hypersurfaces or hypersurfaces which are starshaped with respect to generalized dilations. This paper is the first to study results of this kind for hypersurfaces which are starshaped with respect to general conformal Killing vector fields perturbed by an isometric Killing vector field and our flows allow us to establish isoperimetric inequalities for a much wider class of hypersurfaces. For example, our results apply to hypersurfaces in which are far from being starshaped in the traditional sense, but are starshaped with respect to the conformal Killing vector field composed of a dilation and a rotational vector field. The flow considered in this paper is a novel modification of the mean curvature-type flow first introduced by Guan and Li, which was later generalized by Guan-Li-Wang and Li-Pan.

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Submitted
Sep 26, 2023
Subjects
math.DG, math.AP

Comments: 26 pages, 2 figures

Mean curvature flowVolume-preserving & constrained flowsGeometric inequalities

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 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 Mean curvature type flow and sharp Micheal-Simon inequalities (J. Cui and P. Zhao, 2021)1Cui 2021 Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball (Liangjun Weng and Chao Xia, 2021)6Weng 2021 Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary (Guofang Wang, Liangjun Weng and Chao Xia, 2022)4Wang 2022 A constrained mean curvature flow and Alexandrov-Fenchel inequalities (Xinqun Mei, Guofang Wang and Liangjun Weng, 2022)7Mei 2022 Michael-Simon type inequalities in hyperbolic space ℍⁿ⁺¹ via Brendle-Guan-Li's flows (Jingshi Cui and Peibiao Zhao, 2022)3Cui 2022 The quermassintegral preserving mean curvature flow in the sphere (Esther Cabezas-Rivas and Julian Scheuer, 2022)5Cabezas-Riva 2022 Stability of volume and area preserving mean curvature flow in asymptotic Schwarzschild space (Yaoting Gui, Yuqiao Li and Jun Sun, 2025)8Gui 2025 The quermassintegral inequalities for k-convex spiral-star-shaped domains (Junfang Li, 2026)2Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean curvature type flow and sharp Micheal-Simon inequalities

    J. Cui and P. ZhaoWider flows

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

  2. The quermassintegral inequalities for k-convex spiral-star-shaped domains

    Junfang LiWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows · Shared terms “conformal Killing”, “dilations”, “vector field”

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

    Jingshi Cui and Peibiao ZhaoWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows · Shared terms “Guan”, “Wang”

  4. Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary

    Guofang Wang, Liangjun Weng and Chao XiaWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows · Shared terms “convex hypersurfaces”, “isoperimetric”

  5. The quermassintegral preserving mean curvature flow in the sphere

    Esther Cabezas-Rivas and Julian ScheuerWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows · Shared terms “curvature type”, “convex hypersurfaces”

  6. Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball

    Liangjun Weng and Chao XiaWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows · Shared terms “convex hypersurfaces”

  7. A constrained mean curvature flow and Alexandrov-Fenchel inequalities

    Xinqun Mei, Guofang Wang and Liangjun WengWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows · Shared terms “convex hypersurfaces”

  8. Stability of volume and area preserving mean curvature flow in asymptotic Schwarzschild space

    Yaoting Gui, Yuqiao Li and Jun SunWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows · Shared terms “isoperimetric”

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) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

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.