Equilibrium of Surfaces in a Vertical Force Field

In this paper we study $\varphi$-minimal surfaces in $\mathbb{R}^3$ when the function $\varphi$ is invariant under a two-parametric group of translations. Particularly those which are complete graphs over domains in $\mathbb{R}^2$. We describe a full classification of complete flat embedded $\varphi$-minimal surfaces if $\varphi$ is strictly monotone and characterize rotational $\varphi$-minimal surfaces by its behavior at infinity when $\varphi$ has a quadratic growth.