Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II

Abstract

In this paper we first establish an optimal Sobolev type inequality for hypersurfaces in $\H^n$(see Theorem \ref{mainthm1}). As an application we obtain hyperbolic Alexandrov-Fenchel inequalities for curvature integrals and quermassintegrals. Precisely, we prove a following geometric inequality in the hyperbolic space $\H^n$, which is a hyperbolic Alexandrov-Fenchel inequality, \begin{equation*} \begin{array}{rcl} \ds \int_Σ\s_{2k}\ge \ds\vs C_{n-1}^{2k}ω_{n-1}\left\{\left(\frac{|Σ|}{ω_{n-1}} \right)^\frac 1k + \left(\frac{|Σ|}{ω_{n-1}} \right)^{\frac 1k\frac {n-1-2k}{n-1}} \right\}^k, \end{array} \end{equation*} provided that $\Sigma$ is a horospherical convex, where $2k\le n-1$. Equality holds if and only if $\Sigma$ is a geodesic sphere in $\H^n$. Here $σ_{j}=\s_{j}(κ)$ 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 quermassintegrals in $\H^n$ is as following: $$W_{2k+1}(\Omega )\ge \frac{{\omega }_{n-1}}{n}\sum _{i=0}^k\frac{n-1-2k}{n-1-2k+2i}{C}_k^i(\frac{n{W}_1(\Omega )}{{\omega }_{n-1}}{)}^{\frac{n-1-2k+2i}{n-1}},$$ provided that $Ω\subset\H^n$ is a domain with $\Sigma =\partial \Omega$ horospherical convex, where $2k\le n-1$. Equality holds if and only if $\Sigma$ is a geodesic sphere in $\H^n$. Here $W_r(\Omega )$ is quermassintegrals in integral geometry.

21 pages