Willmore-type inequality in unbounded convex sets

TL;DR

This paper establishes a new Willmore-type inequality for hypersurfaces within unbounded convex sets, relating mean curvature integrals to the set’s asymptotic volume ratio, with equality characterizations and anisotropic extensions.

Contribution

It introduces a novel Willmore-type inequality in unbounded convex sets, including conditions for equality and an anisotropic version, extending classical geometric inequalities.

Findings

Proved a Willmore-type inequality involving the asymptotic volume ratio.

Characterized equality cases as spherical segments and cone structures.

Extended the inequality to anisotropic and capillary hypersurfaces.

Abstract

In this paper we prove the following Willmore-type inequality: On an unbounded closed convex set $K\subset\mathbb{R}^{n+1}$ $(n\ge 2)$, for any embedded hypersurface $\Sigma\subset K$ with boundary $\partial\Sigma\subset \partial K$ satisfying a certain contact angle condition, there holds $$\frac1{n+1}\int_{\Sigma}\vert{H}\vert^n{\rm d}A\ge{\rm AVR}(K)\vert\mathbb{B}^{n+1}\vert.$$ Moreover, equality holds if and only if $\Sigma$ is a part of a sphere and $K\setminus\Omega$ is a part of the solid cone determined by $\Sigma$. Here $\Omega$ is the bounded domain enclosed by $\Sigma$ and $\partial K$, $H$ is the normalized mean curvature of $\Sigma$, and ${\rm AVR}(K)$ is the asymptotic volume ratio of $K$. We also prove an anisotropic version of this Willmore-type inequality. As a special case, we obtain a Willmore-type inequality for anisotropic capillary hypersurfaces in a half-space.