Willmore-type inequality in unbounded convex sets

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 $\hat{I}\pounds \subset K$ with boundary $\partial \hat{I}\pounds \subset \partial K$ satisfying a certain contact angle condition, there holds $$\frac{1}{n+1}{\int }_{\hat{I}}\pounds |H{|}^n\mathrm{d}A\ge \mathrm{AVR}(K)|{\mathbb{B}}^{n+1}|.$$ Moreover, equality holds if and only if $\hat{I}\pounds$ is a part of a sphere and $K\setminus \hat{I}\text{c◯}$ is a part of the solid cone determined by $\hat{I}\pounds$. Here $\hat{I}\text{c◯}$ is the bounded domain enclosed by $\hat{I}\pounds$ and $\partial K$, $H$ is the normalized mean curvature of $\hat{I}\pounds$, and $\mathrm{AVR}(K)$ 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.