Calabi-Bernstein type results for critical points of a weighted area functional in ${\mathbb{R}}^3$ and ${\mathbb{L}}^3$

In this paper we prove some Calabi-Bernstein type and non-existence results concerning complete $[\phi ,{\vec{e}}_3]$-minimal surfaces in ${\mathbb{R}}^3$ whose Gauss maps lie on compacts subsets of open hemispheres of ${\mathbb{S}}^2$. We also give a general non-existence result for complete spacelike $[\phi ,{\vec{e}}_3]$-maximal surfaces in ${\mathbb{L}}^3$ and, in particular, we obtain a Calabi-Bernstein type result when $\dot{\phi }$ is bounded.