Dual quadratic differentials and entire minimal graphs in Heisenberg space

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.

19 pages

References in corpus (3)

• General curvature estimates for stable H-surfaces in 3-manifolds and applications
• A solution to the Bernstein problem in the three-dimensional Heisenberg group via loop groups
• On the conformal duality between constant mean curvature surfaces in $\mathbb{E}(\kappa ,\tau )$ and $\mathbb{L}(\kappa ,\tau )$

Cited by in corpus (3)

• Isoparametric surfaces in $\mathbb{E}(\kappa ,\tau )$-spaces
• A duality for prescribed mean curvature graphs in Riemannian and Lorentzian Killing submersions
• Conjugate Plateau constructions in product spaces