Graphic Bernstein Results in Curved Pseudo-Riemannian Manifolds

Abstract

We generalize a Bernstein-type result due to Albujer and Al\'ias, for maximal surfaces in a curved Lorentzian product 3-manifold of the form $\Sigma_1\times \mathbb{R}$, to higher dimension and codimension. We consider $M$ a complete spacelike graphic submanifold with parallel mean curvature, defined by a map $f: \Sigma_1\to \Sigma_2$ between two Riemannian manifolds $(\Sigma_1^m, g_1)$ and $(\Sigma^n_2, g_2)$ of sectional curvatures $K_1$ and $K_2$, respectively. We take on $\Sigma_1\times \Sigma_2$ the pseudo-Riemannian product metric $g_1-g_2$. Under the curvature conditions, $\mathrm{Ricci}_1 \geq 0$ and $K_1\geq K_2$, 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 $\mathrm{Ricci}_1(p)>0$ at some point. For bounded $K_1$, $K_2$ and hyperbolic angle $\theta$, we conclude $M$ must be maximal. If $M$ is a maximal surface and $K_1\geq K_2^+$, we show $M$ is totally geodesic with no need for further assumptions. Furthermore, $M$ is a slice if at some point $p\in \Sigma_1$, $K_1(p)> 0$, and if $\Sigma_1$ is flat and $K_2<0$ at some point $f(p)$, then the image of $f$ lies on a geodesic of $\Sigma_2$.