Weighted Alexandrov–Fenchel inequalities in hyperbolic space and a conjecture of Ge, Wang, and Wu

Abstract

We consider a conjecture made by Ge, Wang, and Wu regarding weighted Alexandrov–Fenchel inequalities for horospherically convex hypersurfaces in hyperbolic space (a bound, for some physically motivated weight function, of the weighted integral of the k k th mean curvature in terms of the area of the hypersurface). We prove an inequality very similar to the conjectured one. Moreover, when k k is zero and the ambient space has dimension three, we give a counterexample to the conjectured inequality.