Graphic Bernstein Results in Curved Pseudo-riemannian Manifolds

We generalize a Bernstein-type result due to Albujer and Alı́as, 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 (Σ1 ,g1) and (Σ n 2,g2) of sectional curvatures K1 and K2, respectively. We take on Σ1×Σ2 the pseudoRiemannian 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 M must be maximal. If M is a maximal surface and K1 ≥ K + 2 , 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. 0 MSC 2000: Primary: 53C21, 53C42, 53C50