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New general results of non-existence and rigidity of spacelike submanifoldsimmersed in a spacetime, whose mean curvature is a time-oriented causal vectorfield, are given. These results hold for a wide class of spacetimes whichincludes globally hyperbolic, stationary, conformally stationary and pp-wavespacetimes, among others. Moreover, applications to the Cauchy problem inGeneral Relativity, are presented. Finally, in the case of hypersurfaces, wealso obtain significant consequences in Geometrical Analysis, solving newCalabi-Bernstein and Dirichlet problems on a Riemannian manifold.

Rigidity and non-existence of spacelike submanifolds with causal mean  curvature vector field in spacetimes and the Cauchy problem in General  Relativity