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 $\mathbb {H}^n$:

$$\begin{aligned} \int _{\Sigma } p_{2k}\ge \omega _{n-1}\left[ \left( \frac{|\Sigma |}{\omega _{n-1}}\right) ^\frac{1}{k} +\left( \frac{|\Sigma |}{\omega _{n-1}}\right) ^{\frac{1}{k}\frac{n-1-2k}{n-1}}\right] ^k, \end{aligned}$$

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 $\mathbb {H}^n$:

$$\begin{aligned} W_{2k+1}(\Omega )\ge \frac{\omega _{n-1}}{n}\sum _{i=0}^{k}\frac{n-1-2k}{n-1-2i} C_k^i\left( \frac{|\Sigma |}{\omega _{n-1}}\right) ^\frac{n-1-2i}{n-1}, \end{aligned}$$

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 were previously proved by Ge et al. (J Differ Geom 98:237–260, 2014) under the stronger condition that $\Sigma$ is horospherical convex.