Studying riemannian immersions into semi-riemannian spaces via parabolicity, Liouville type results and other maximum principles

Abstract

This thesis studies the geometry of complete Riemannian submanifolds immersed in certain semi-Riemannian spaces via parabolicity criteria related to modified ChengYau’s operators and to a linearized differential operator which can be regarded as a natural extension of the standard Laplacian, via generalization of a Liouville-type result and versions of maximum principle. In this regard, via parabolicity criteria and from appropriate Simons type formulas concerning spacelike submanifolds immersed with parallel normalized mean curvature vector in Einstein Manifolds we prove new characterization results. In the case of submanifolds of semi-Riemannian warped products, under standard convergence conditions and appropriated constraints on the higher order mean curvatures, we also obtain uniqueness and nonexistence results via parabolicity and p-integrability criteria, for p ≥ 1, generalization of a Liouville-type result, a version of maximum principle at infinity for vector fields and a maximum principle related to polynomial volume growth. Applications are also presented to cases in which the ambient space is either an Einstein manifold, the Steady State models, Schwarzschild and Reeissner-Nordström spaces, and a particular investigation of entire graphs constructed over the fiber of the ambient space.