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 $[\varphi,\vec{e}_{3}]$-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 $[\varphi,\vec{e}_{3}]$-maximal surfaces in $\mathbb{L}^{3}$ and, in particular, we obtain a Calabi-Bernstein type result when $\dot{\varphi}$ is bounded.