Stochastically Complete, Parabolic and L1-Liouville Spacelike Submanifolds with Parallel Mean Curvature Vector

We deal with n -dimensional spacelike submanifolds immersed with parallel mean curvature vector h in a pseudo-Riemannian space form L q n + p ( c ) of index 1 ≤ q ≤ p and constant sectional curvature c ∈{− 1,0,1}. Considering the cases when h is either spacelike or timelike, we are able to prove that such a spacelike submanifold is either totally umbilical or it holds a lower estimate for the supremum of the norm of its traceless second fundamental form, occurring equality if the spacelike submanifold is pseudo-umbilical and its principal curvatures are constant. In our approach, we use three main core concepts: Stochastic completeness, parabolicity and L 1 -Liouville property.