Desigualdades tipo Alexandrov-Fenchel para hipersuperfícies da esfera e do espaço hiperbólico.

Abstract

This thesis is divided in two parts. In the first, we find a monotone quantity along the inverse mean curvature flow and use it to prove an Alexandrov–Fenchel-type inequality for strictly convex hypersurfaces in the sphere. In the second part, we consider an inequality conjectured by Ge, Wang and Wu in 2015 for hypersurfaces in the hyperbolic space. Using a geometric flow, which we call the support function flow, and a monotone quantity along of this flow, we prove an inequality similar to the one that was conjectured. Moreover, when the dimension of the ambient manifold is three, we show that the inequality that was conjectured is false.