Capillary quermassintegral inequalities in the unit ball

TL;DR

This paper introduces a new convexity notion for hypersurfaces in the unit ball, proves curvature flow convergence, and establishes quermassintegral inequalities for a broad class of curvature functions.

Contribution

It develops the concept of θ-horocap-convexity, proves curvature flow convergence, and derives full quermassintegral inequalities in this setting.

Findings

Curvature flow converges for θ-horocap-convex hypersurfaces.

Full set of quermassintegral inequalities established for these hypersurfaces.

New methods characterize equality cases in geometric inequalities.

Abstract

This paper is about hypersurfaces with boundary lying in the Euclidean unit ball, which meet the unit sphere at a fixed angle $\theta\in(0,\frac{\pi}{2}]$. Such hypersurfaces are called $\theta$-capillary hypersurfaces and for those we introduce a new notion of convexity, which we call $\theta$-horocap-convexity. For such hypersurfaces, we prove the convergence of a curvature flow of Guan/Li type with capillary boundary. Remarkably, we prove this result for a class of curvature functions which include all quotients of symmetric polynomials and, as a consequence, we obtain the full set of quermassintegral inequalities in the $\theta$-horocap-convex case. In the strictly horocap-convex setting, we employ the flow to prove the geometric inequalities, while for the horocap-convex case and the characterization of the equality case, we develop new arguments which are interesting in their own right.