Extensions of Calabi's correspondence between minimal surfaces and maximal surfaces

TL;DR

This survey explores extensions of Calabi's duality between minimal surfaces in Euclidean space and maximal surfaces in Lorentz space, highlighting geometric applications of the Poincaré Lemma to constant mean curvature equations.

Contribution

The paper introduces two new extensions of Calabi's correspondence, broadening the understanding of the duality between minimal and maximal surfaces.

Findings

Constructed two extensions of Calabi's correspondence

Illustrated geometric applications of the Poincaré Lemma

Enhanced the theoretical framework connecting minimal and maximal surfaces

Abstract

The main goal of this survey is to illustrate geometric applications of the Poincar\'{e} Lemma to constant mean curvature equations. In 1970, Calabi introduced the duality between minimal graphs in three dimensional Euclidean space and maximal graphs in three dimensional Lorentz space. We construct two extensions of Calabi's correspondence.