Geometric inequalities for hypersurfaces with nonnegative sectional curvature in hyperbolic space

In this article, we will use inverse mean curvature flow to establish an optimal Sobolev-type inequality for hypersurfaces Σ with nonnegative sectional curvature in H ⁿ . As an application, we prove the hyperbolic Alexandrov–Fenchel inequalities for hypersurfaces with nonnegative sectional curvature in H ⁿ : ∫Σp2k≥ωn-1[(|Σ|ωn-1)1k+(|Σ|ωn-1)1kn-1-2kn-1]k,where p _i is the normalized i-th mean curvature. Equality holds if and only if Σ is a geodesic sphere in H ⁿ . For a domain Ω ⊂ H ⁿ with Σ = ∂Ω having nonnegative sectional curvature, we prove an optimal inequality for quermassintegral in H ⁿ : W2k+1(Ω)≥ωn-1n∑i=0kn-1-2kn-1-2iCki(|Σ|ωn-1)n-1-2in-1,where W _i (Ω) is the i-th quermassintegral in integral geometry. Equality holds if and only if Σ is a geodesic sphere in H ⁿ . All these inequalities were previously proved by Ge et al. (J Differ Geom 98:237–260, 2014) under the stronger condition that Σ is horospherical convex.