A Calabi's Type Correspondence

Calabi observed that there is a natural correspondence between the solutions of the minimal surface equation in $\mathbb{R}^3$ with those of the maximal spacelike surface equation in $\mathbb{L}^3$. We are going to show how this correspondence can be extended to the family of $\varphi$-minimal graphs in $\mathbb{R}^3$ when the function $\varphi$ is invariant under a two-parametric group of translations. We give also applications in the study and description of new examples.