Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II

Abstract

In this paper we first establish an optimal Sobolev-type inequality for hypersurfaces in $\mathbb{H}^n$ (see Theorem 1.1). As an application we obtain hyperbolic Alexandrov-Fenchel inequalities for curvature integrals and quermassintegrals. More precisely, we prove a geometric inequality in the hyperbolic space $\mathbb{H}^n$, which is a hyperbolic Alexandrov-Fenchel inequality,

$$ \begin{array}{rcl} \int_\Sigma \sigma_{2k}\ge C_{n-1}^{2k}\omega_{n-1}\left\{ \left( \frac{|\Sigma|}{\omega_{n-1}} \right)^\frac 1k +\left( \frac{|\Sigma|}{\omega_{n-1}} \right)^{\frac 1k\frac {n-1-2k}{n-1}} \right\}^k, \end{array} $$

when $\Sigma$ is a horospherical convex and $2k\leq n-1$. Equality holds if and only if $\Sigma$ is a geodesic sphere in $\mathbb{H}^n$. Here $\sigma_{j}=\sigma_{j}(\kappa)$ is the $j$th mean curvature and $\kappa=(\kappa_1,\kappa_2,\cdots, \kappa_{n-1})$ is the set of the principal curvatures of $\Sigma$. Also, an optimal inequality for quermassintegral in $\mathbb{H}^n$ is

$$W_{2k+1}(\Omega)\geq\frac {\omega_{n-1}}{n}\sum_{i=0}^k\frac{n-1-2k}{n-1-2k+2i}\,C_k^i\bigg(\frac{nW_1(\Omega)}{\omega_{n-1}}\bigg)^{\frac{n-1-2k+2i}{n-1}},$$

provided that $\Omega\subset\mathbb{H}^n$ is a domain with $\Sigma=\partial\Omega$ horospherical convex, where $2k\leq n-1$. Equality holds if and only if $\Sigma$ is a geodesic sphere in $\mathbb{H}^n$. Here $W_r(\Omega)$ is quermassintegrals in integral geometry.