Dual quadratic differentials and entire minimal graphs in Heisenberg space

TL;DR

This paper introduces dual quadratic differentials for spacelike surfaces in Lorentzian spaces, providing new proofs and geometric insights into entire minimal graphs in Heisenberg space, including their curvature properties.

Contribution

It defines dual quadratic differentials in Lorentzian spaces, offers a concise proof of the Bernstein problem in Heisenberg space, and characterizes entire minimal graphs' curvature.

Findings

Short proof of the Bernstein problem in Heisenberg space

Geometric description of entire graphs sharing the same differential

Entire minimal graphs in Heisenberg space have negative Gauss curvature

Abstract

We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces $\mathbb{L}(\kappa,\tau)$ with isometry group of dimension 4, which are dual to the Abresch-Rosenberg differentials in the Riemannian counterparts $\mathbb{E}(\kappa,\tau)$, and obtain some consequences. On the one hand, we give a very short proof of the Bernstein problem in Heisenberg space, and provide a geometric description of the family of entire graphs sharing the same differential in terms of a 2-parameter conformal deformation. On the other hand, we prove that entire minimal graphs in Heisenberg space have negative Gauss curvature.