## Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space.(English)Zbl 1432.53134

Summary: 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 $$\mathbb {H}^n$$: $\int _{\Sigma } p_{2k}\geq \omega _{n-1}\left[ \left( \frac{|\Sigma |}{\omega _{n-1}}\right) ^\frac{1}{k} +\left( \frac{|\Sigma |}{\omega _{n-1}}\right) ^{\frac{1}{k}\frac{n-1-2k}{n-1}}\right] ^k,$ 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 $$\mathbb {H}^n$$: $W_{2k+1}(\Omega )\geq \frac{\omega _{n-1}}{n}\sum _{i=0}^{k}\frac{n-1-2k}{n-1-2i} C_k^i\left( \frac{|\Sigma |}{\omega _{n-1}}\right) ^\frac{n-1-2i}{n-1},$ 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 Y. Ge et al. [J. Differ. Geom. 98, No. 2, 237–260 (2014; Zbl 1301.53077)] under the stronger condition that $$\Sigma$$ is horospherical convex.

### MSC:

 53E10 Flows related to mean curvature 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 52A40 Inequalities and extremum problems involving convexity in convex geometry

Zbl 1301.53077
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### References:

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