A Calabi's Type Correspondence

Abstract

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 $\phi$-minimal graphs in ${\mathbb{R}}^3$ when the function $\phi$ is invariant under a two-parametric group of translations. We give also applications in the study and description of new examples.

21 pages, 27 figures