Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space

Abstract

In this article, we will use inverse mean curvature flow to establish an optimal Sobolev-type inequality for hypersurfaces $\Sigma$ with nonnegative sectional curvature in $\mathbb{H}^n$. As an application, we prove the hyperbolic Alexandrov-Fenchel inequalities for hypersurfaces with nonnegative sectional curvature in $\mathbb{H}^n$: \begin{align*} \int_{\Sigma} p_{2k}\geq \omega_{n-1}\left[\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^\frac{1}{k}+\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^{\frac{1}{k}\frac{n-1-2k}{n-1}}\right]^k, \end{align*} where $p_i$ is the normalized $i$-th mean curvature. Equality holds if and only if $\Sigma$ is a geodesic sphere in $\mathbb{H}^n$. For a domain $\Omega\subset \mathbb{H}^n$ with $\Sigma=\partial \Omega$ having nonnegative sectional curvature, we prove an optimal inequality for quermassintegral in $\mathbb{H}^n$: \begin{align*} W_{2k+1}(\Omega)\geq \frac{\omega_{n-1}}{n}\sum_{i=0}^{k}\frac{n-1-2k}{n-1-2i}C_k^i\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^\frac{n-1-2i}{n-1}, \end{align*} where $W_i(\Omega)$ is the $i$-th quermassintegral in integral geometry. Equality holds if and only if $\Sigma$ is a geodesic sphere in $\mathbb{H}^n$. All these inequalities was previously proved by Ge, Wang and Wu \cite{Ge-Wang-Wu2014} under the stronger condition that $\Sigma$ is horospherical convex.