# A Calabi's Type Correspondence

## TL;DR

This paper extends Calabi's correspondence between minimal and maximal surfaces to a broader class of -minimal graphs invariant under translations, providing new examples and applications in differential geometry.

## Contribution

It generalizes Calabi's correspondence to -minimal graphs with translation invariance, introducing new examples and applications.

## Key findings

- Extended Calabi's correspondence to -minimal graphs
- Constructed new examples of -minimal surfaces
- Provided applications in geometric analysis

## 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 $\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.