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Hyperbolic Alexandrov-Fenchel quermassintegral inequalities II

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 $Σ$ is a horospherical convex, where $2k\leq n-1$. Equality holds if and only if $Σ$ is a geodesic sphere in $\H^n$. Here $σ_{j}=\s_{j}(κ)$ is the $j$-th mean curvature and $κ=(κ_1,κ_2,\cdots, κ_{n-1})$ is the set of the principal curvatures of $Σ$. Also, an optimal inequality for quermassintegrals in $\H^n$ is as following: $$ W_{2k+1}(Ω)\geq\frac {ω_{n-1}}{n}\sum_{i=0}^k\frac{n-1-2k}{n-1-2k+2i}\,C_k^i\bigg(\frac{nW_1(Ω)}{ω_{n-1}}\bigg)^{\frac{n-1-2k+2i}{n-1}}, $$ provided that $Ω\subset\H^n$ is a domain with $Σ=\partialΩ$ horospherical convex, where $2k\leq n-1$. Equality holds if and only if $Σ$ is a geodesic sphere in $\H^n$. Here $W_r(Ω)$ is quermassintegrals in integral geometry.