Straight ruled surfaces in the Heisenberg group

TL;DR

This paper extends the understanding of minimal surfaces in the Heisenberg group by showing they are locally straight ruled surfaces and contactomorphic to the complex plane, generalizing previous results.

Contribution

It generalizes a known result by proving all horizontally minimal surfaces in the Heisenberg group are locally straight ruled surfaces and contactomorphic to the complex plane.

Findings

Horizontally minimal surfaces are locally straight ruled surfaces.

Such surfaces are locally contactomorphic to the complex plane.

The result generalizes previous work by Garofalo and Pauls.

Abstract

We generalise a result of Garofalo and Pauls: a horizontally minimal smooth surface embedded in the Heisenberg group is locally a (straight) ruled surface, i.e. it consists of straight lines tangent to a horizontal vector field along a smooth curve. We show additionally that any horizontally minimal surface is locally contactomorphic to the complex plane.