Comportamento assintótico do fluxo pelo inverso da curvatura média em ambientes subestáticos

Abstract

In this thesis, we study the behaviour of the inverse mean curvature fl ow (IMCF) in the AdS-Reissner-Nordström space and in its particular cases (hyperbolic space and AdS-Schwarzschild space). We show that, in these spaces, the behaviour of the IMCF is quite different from its behaviour when the ambient is the Euclidean space. More precisely, using a method similar to the one used by Hung and Wang, we show the existence of a hypersurface in the Ads-Schwarzschild space whose induced metric of its evolution by the IMCF, normalized by the appropriate power of the area, does not converge to a space form. Moreover, we obtain an inequality involving geometric quantities and some results about the asymptotic behaviour of the IMCF. Finally, we obtain two weighted Alexandrov-Fenchel-type inequalities similar to an inequality proved by Ge, Wang and Wu; the fi rst one is an inequality for hypersurfaces in hyperbolic space and involves the even order mean curvatures, while the second one is an inequality for hypersurfaces in Euclidean space involving the odd order mean curvatures.