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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

Citations:

Zbl 1301.53077
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References:

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