Caracterização de hipersuperfície com curvatura r-média constante

Abstract

This thesis aims to present a proof of a theorem of Sebastián Montiel and Antonio Ros which characterizes the compact hypersurfaces, without boundary, with constant r-mean curvature embedded in the space forms. This theorem generalizes the classical theorem of Alexandrov and states that The only compact, without boundary, hypersurfaces with constant r-mean curvature for some r = 1, . . . , n, embedded in R n+1 , in the open hemisphere of the Euclidean sphere S n+1 or in the hyperbolic space Hn+1, are the geodesic hyperspheres. We remark that, despite we follow the ideas of Montiel and Ros, we give a new approach for some steps of the proof presented here.