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 ${\Sigma }_1\times \mathbb{R}$, 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 $({\Sigma }_2^n,g_2)$ 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, ${\mathrm{Ricci}}_1\ge 0$ and $K_1\ge K_2$, 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 $K_1\ge K_2^{+}$, 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 ${\Sigma }_2$.

Accepted for publication in the Journal of Geometry and Physics. Final version: Some simplifications, improvements and reorganization. In version 3, we replace the condition $K_1\ge 0$ by the weaker condition $Ricc{i}_1\ge 0$. The proofs are essentially the same

References in corpus (2)

• Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces
• Bernstein Theorems for Space-like Graphs with Parallel Mean Curvature and Controlled Growth