Alexandrov-Fenchel type inequalities for convex hypersurfaces in hyperbolic space and in sphere

TL;DR

This paper establishes Alexandrov-Fenchel type inequalities for convex hypersurfaces in hyperbolic space and spheres, using isoperimetric inequalities and inverse mean curvature flow, with results on rigidity and optimal Sobolev inequalities.

Contribution

It derives new Alexandrov-Fenchel inequalities in hyperbolic space and spheres, and proves an optimal Sobolev inequality for convex hypersurfaces in the sphere.

Findings

Derived Alexandrov-Fenchel inequalities in hyperbolic space and sphere.

Established rigidity results in the spherical case.

Proved an optimal Sobolev inequality for convex hypersurfaces in the sphere.

Abstract

In this paper, firstly, inspired by Nat\'{a}rio's recent work \cite{Na}, we use the isoperimetric inequality to derive some Alexandrov-Fenchel type inequalities for closed convex hypersurfaces in the hyperbolic space $\H^{n+1}$ and in the sphere $\SS^{n+1}$. We also get the rigidity in the spherical case. Secondly, we use the inverse mean curvature flow in sphere \cite{gerh,Mak-Sch} to prove an optimal Sobolev type inequality for closed convex hypersurfaces in the sphere.