Minimal null scrolls in the three-dimensional Heisenberg group

TL;DR

This paper characterizes timelike minimal surfaces in the 3D Heisenberg group with a specific Lorentzian metric, linking them to null curves and affine null lines, and provides methods to construct such surfaces with given curvatures.

Contribution

It introduces a new characterization of minimal null scrolls in the Heisenberg group using Abresch-Rosenberg differentials and null curve constructions.

Findings

Timelike minimal surfaces are characterized by null curve multiplications.

Construction methods for surfaces with prescribed null curve curvatures.

Explicit description of surfaces via null lines and curves.

Abstract

In the three-dimensional Heisenberg group equipped with a certain left invariant Lorentzian metric, timelike minimal surfaces which have the Abresch-Rosenberg differentials with vanishing multiplication of the coefficient function and its para-complex conjugate are characterized as the surfaces defined by the multiplication of null curves and affine null lines. Moreover, the constructions of these surfaces with prescribed curvatures of null curves or prescribed null lines are given.