Aplicaciones del principio del máximo generalizado de omori-yau al estudio de la geometría global de hipersuperficies en espacios de curvatura constante

Abstract

ABSTRACT The goal of this work is to show the evolution of the maximum principle and several applications of this to geometric problems. In this sense, we study the behavior of the scalar curvature S of hypersurfaces immersed with constant mean curvature into a Riemannian space form, under non-compactness's hypotheses as: the completeness and the stochastic completeness, obtaining a sharp estimate for the infimum of S. Moreover, we study these hypersurfaces with the conditions of two principal curvatures and satisfying the Omori-Yau maximum principle, deriving a sharp estimate for the supremum of S. Finally, we establish a weak maximum principle of differential operator L, introduced by Cheng and Yau [19] for study of complete hypersurfaces with constant scalar curvature, and give an application where we estimate the infimum of the mean curvature of these hypersurfaces . The results of this work are collected in the papers [5], [6] and [7]. KEY WORDS: Omori-Yau maximum principle, constant mean curvature, scalar curvature, traceless second fundamental form, first Newton tensor, complete, stochastic completeness, parabolicity and two distinct principal curvatures.