Graphic Bernstein Results in Curved Pseudo-Riemannian Manifolds

TL;DR

This paper extends Bernstein-type results for maximal surfaces in curved Lorentzian product manifolds to higher dimensions and codimensions, establishing conditions under which such submanifolds are totally geodesic or slices.

Contribution

It generalizes previous results to higher dimensions and codimensions, providing new curvature conditions and integrability criteria for totally geodesic and maximal submanifolds.

Findings

Submanifolds are totally geodesic under certain curvature and integrability conditions.

Maximal submanifolds with bounded curvatures are necessarily slices.

Image of the defining map lies on a geodesic under specific curvature conditions.

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 $\Sigma_2$.