Graphic Bernstein results in curved pseudo-Riemannian manifolds

We generalize a Bernstein-type result due to Albujer and Alías, for maximal surfaces in a curved Lorentzian product 3-manifold of the form Σ1×R, to higher dimension and codimension. We consider M a complete spacelike graphic submanifold with parallel mean curvature, defined by a map f:Σ1→Σ2 between two Riemannian manifolds (Σ1m,g1) and (Σ2n,g2) of sectional curvatures K1 and K2, respectively. We take on Σ1×Σ2 the pseudo-Riemannian product metric g1−g2. Under the curvature conditions, Ricci1≥0 and K1≥K2, we prove that, if the second fundamental form of M satisfies an integrability condition, then M is totally geodesic, and it is a slice if Ricci1(p)>0 at some point. For bounded K1, K2 and hyperbolic angle θ, we conclude that M must be maximal. If M is a maximal surface and K1≥K2+, we show M is totally geodesic with no need for further assumptions. Furthermore, M is a slice if at some point p∈Σ1, K1(p)>0, and if Σ1 is flat and K2<0 at some point f(p), then the image of f lies on a geodesic of Σ2.