Ruled minimal surfaces in the three dimensional Heisenberg group

TL;DR

This paper classifies all ruled minimal surfaces in the three-dimensional Heisenberg group, showing they are limited to planes, helicoids, and hyperbolic paraboloids, and examines their mean curvature properties under different metrics.

Contribution

It provides a complete classification of ruled minimal surfaces in the Heisenberg group and analyzes their mean curvature in Riemannian and Lorentzian settings.

Findings

Planes, helicoids, and hyperbolic paraboloids are the only ruled minimal surfaces in the Heisenberg group.

These surfaces have zero mean curvature under both Riemannian and Lorentzian metrics.

The classification extends understanding of minimal surfaces in sub-Riemannian geometry.

Abstract

It is shown that parts of planes, helicoids and hyperbolic paraboloids are the only minimal surfaces ruled by geodesics in the three dimensional Riemannian Heisenberg group. It is also shown that they are the only surfaces in the three dimensional Heisenberg group whose mean curvature is zero with respect to both of the standard Riemannian metric and the standard Lorentzian metric.