Paper · Nov 14, 2018

Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball

Julian Scheuer, Guofang Wang, Chao Xia

math.DGWider flowsarXiv:1811.05776
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Abstract

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the -dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for we obtain a Minkowski-type inequality and for we obtain an optimal Willmore-type inequality. To prove these estimates, we employ a specifically designed locally constrained inverse harmonic mean curvature flow with free boundary.

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Submitted
Nov 14, 2018
Subjects
math.DG, math.AP
Journal
J. Differ. Geom. 120 (2022), no. 2, p. 345-373
DOI
10.4310/jdg/1645207496

Comments: 21 pages. Comments welcome

Mean curvature flowVolume-preserving & constrained flowsFree boundary & capillaryGeometric inequalities

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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 A geometric inequality for convex free boundary hypersurfaces in the unit ball (Ben Lambert and Julian Scheuer, 2016)5Lambert 2016 Locally constrained inverse curvature flows (Julian Scheuer and Chao Xia, 2017)6Scheuer 2017 Guan-Li type mean curvature flow for free boundary hypersurfaces in a ball (Guofang Wang and Chao Xia, 2019)3Wang 2019 Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in a ball (Liangjun Weng and Chao Xia, 2021)2Weng 2021 Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary (Guofang Wang, Liangjun Weng and Chao Xia, 2022)1Wang 2022 A constrained mean curvature flow and Alexandrov-Fenchel inequalities (Xinqun Mei, Guofang Wang and Liangjun Weng, 2022)4Mei 2022 Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space (Xinqun Mei and Liangjun Weng, 2024)7Mei 2024 Capillary quermassintegral inequalities in the unit ball (Shujing Pan and Julian Scheuer, 2026)8Pan 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary

    Guofang Wang, Liangjun Weng and Chao XiaWider flows

    Topics Free boundary & capillary, Geometric inequalities, Volume-preserving & constrained flows · Same authors Guofang Wang, Chao Xia · Shared terms “Fenchel inequalities”, “Alexandrov Fenchel”, “convex hypersurfaces”

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

    Liangjun Weng and Chao XiaWider flows

    Topics Free boundary & capillary, Geometric inequalities, Volume-preserving & constrained flows · Same author Chao Xia · Shared terms “Fenchel inequalities”, “Alexandrov Fenchel”, “convex hypersurfaces”

  3. Guan-Li type mean curvature flow for free boundary hypersurfaces in a ball

    Guofang Wang and Chao XiaWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows · Same authors Guofang Wang, Chao Xia · Shared terms “boundary hypersurfaces”

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

    Xinqun Mei, Guofang Wang and Liangjun WengWider flows

    Topics Free boundary & capillary, Geometric inequalities, Volume-preserving & constrained flows · Same author Guofang Wang · Shared terms “Fenchel inequalities”, “Alexandrov Fenchel”, “convex hypersurfaces”

  5. A geometric inequality for convex free boundary hypersurfaces in the unit ball

    Ben Lambert and Julian ScheuerWider flows

    Topics Free boundary & capillary, Geometric inequalities · Same author Julian Scheuer · Shared terms “boundary hypersurfaces”, “convex free”, “unit ball”

  6. Locally constrained inverse curvature flows

    Julian Scheuer and Chao XiaWider flows

    Topics Geometric inequalities, Volume-preserving & constrained flows · Same authors Julian Scheuer, Chao Xia · Shared terms “constrained inverse”, “type inequality”, “locally constrained”

  7. Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space

    Xinqun Mei and Liangjun WengWider flows

    Topics Free boundary & capillary, Geometric inequalities, Volume-preserving & constrained flows · Shared terms “Fenchel inequalities”, “Alexandrov Fenchel”, “isoperimetric type”

  8. Capillary quermassintegral inequalities in the unit ball

    Shujing Pan and Julian ScheuerWider flows

    Topics Free boundary & capillary, Geometric inequalities · Same author Julian Scheuer · Shared terms “unit ball”, “Euclidean unit”, “quermassintegral”

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.

Foundational paper in the Free boundary & capillary guide: Scheuer–Wang–Xia: Alexandrov–Fenchel inequalities for convex free-boundary hypersurfaces in a ball.

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) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989) 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.