Paper · Jun 20, 2016

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

Ben Lambert, Julian Scheuer

math.DGWider flowsarXiv:1606.06138
View PDF View HTML arXiv page

As on arXiv

Abstract

We use the inverse mean curvature flow with a free boundary perpendicular to the sphere to prove a geometric inequality involving the Willmore energy for convex hypersurfaces of dimension with boundary on the sphere.

View PDF View HTML arXiv page
Submitted
Jun 20, 2016
Subjects
math.DG
Journal
Proc. Am. Math. Soc. 145, no. 9, p. 4009--4020, (2017)
DOI
10.1090/proc/13516

Comments: 9 pages. Comments are welcome

Mean curvature flowOther curvature flowsInverse mean curvature flowFree boundary & capillaryGeometric 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 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 Willmore inequality on hypersurfaces in hyperbolic space (Yingxiang Hu, 2016)5Hu 2016 Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball (Julian Scheuer, Guofang Wang and Chao Xia, 2018)1Scheuer 2018 Harmonic mean curvature flow and geometric inequalities (Ben Andrews, Yingxiang Hu and Haizhong Li, 2019)8Andrews 2019 Guan-Li type mean curvature flow for free boundary hypersurfaces in a ball (Guofang Wang and Chao Xia, 2019)6Wang 2019 Inverse mean curvature flow with a free boundary in hyperbolic space (Xiaoxiang Chai, 2022)4Chai 2022 A capillary problem for spacelike mean curvature flow in a cone of Minkowski space (Wilhelm Klingenberg, Ben Lambert and Julian Scheuer, 2023)7Klingenberg 2023 Inverse curvature flows for capillary hypersurfaces in the unit ball (Shujing Pan and Bo Yang, 2025)3Pan 2025 Capillary quermassintegral inequalities in the unit ball (Shujing Pan and Julian Scheuer, 2026)2Pan 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 with free boundary in a ball

    Julian Scheuer, Guofang Wang and Chao XiaWider flows

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

  2. Capillary quermassintegral inequalities in the unit ball

    Shujing Pan and Julian ScheuerWider flows

    Topics Free boundary & capillary, Geometric inequalities, Other curvature flows · Same author Julian Scheuer · Shared terms “unit ball”

  3. Inverse curvature flows for capillary hypersurfaces in the unit ball

    Shujing Pan and Bo YangWider flows

    Topics Free boundary & capillary, Geometric inequalities, Inverse mean curvature flow · Shared terms “unit ball”, “convex hypersurfaces”

  4. Inverse mean curvature flow with a free boundary in hyperbolic space

    Xiaoxiang ChaiWider flows

    Topics Free boundary & capillary, Inverse mean curvature flow, Other curvature flows · Shared terms “Willmore”, “convex hypersurfaces”

  5. Willmore inequality on hypersurfaces in hyperbolic space

    Yingxiang HuWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “Willmore”, “convex hypersurfaces”

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

    Guofang Wang and Chao XiaWider flows

    Topics Free boundary & capillary · Shared terms “boundary hypersurfaces”

  7. A capillary problem for spacelike mean curvature flow in a cone of Minkowski space

    Wilhelm Klingenberg, Ben Lambert and Julian ScheuerWider flows

    Topics Free boundary & capillary · Same authors Ben Lambert, Julian Scheuer

  8. Harmonic mean curvature flow and geometric inequalities

    Ben Andrews, Yingxiang Hu and Haizhong LiWider flows

    Topics Geometric inequalities, Inverse mean curvature flow, Other curvature flows · Shared terms “convex hypersurfaces”

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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) 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)

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.