Graphic Bernstein results in curved pseudo-Riemannian manifolds

Abstract

We generalize a Bernstein-type result due to Albujer and Alías, for maximal surfaces in a curved Lorentzian product 3-manifold of the form Σ 1 × R , to higher dimension and codimension. We consider M a complete spacelike graphic submanifold with parallel mean curvature, defined by a map f : Σ 1 → Σ 2 between two Riemannian manifolds ( Σ 1 m , g 1 ) and ( Σ 2 n , g 2 ) of sectional curvatures K 1 and K 2 , respectively. We take on Σ 1 × Σ 2 the pseudo-Riemannian product metric g 1 − g 2 . Under the curvature conditions, Ricci 1 ≥ 0 and K 1 ≥ K 2 , 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 Ricci 1 ( p ) > 0 at some point. For bounded K 1 , K 2 and hyperbolic angle θ , we conclude that M must be maximal. If M is a maximal surface and K 1 ≥ K 2 + , we show M is totally geodesic with no need for further assumptions. Furthermore, M is a slice if at some point p ∈ Σ 1 , K 1 ( p ) > 0 , and if Σ 1 is flat and K 2 < 0 at some point f ( p ) , then the image of f lies on a geodesic of Σ 2 .