Minimal cylinders in the three-dimensional Heisenberg group

TL;DR

This paper investigates minimal cylinders in the three-dimensional Heisenberg group using a loop group method, characterizing all non-vertical minimal cylinders via pairs of closed plane curves with equal signed area, and also constructs spacelike CMC cylinders.

Contribution

It introduces a novel characterization of non-vertical minimal cylinders in Nil_3 through pairs of closed plane curves with equal signed area, using a generalized Weierstrass representation.

Findings

All non-vertical minimal cylinders are characterized by pairs of closed plane curves with equal signed area.

The method also produces spacelike CMC cylinders as a byproduct.

Provides a comprehensive classification of minimal cylinders in Nil_3.

Abstract

We study minimal cylinders in the three-dimensional Heisenberg group ${\rm Nil}_3$ using the generalized Weierstrass type representation, the so-called loop group method. We characterize all non-vertical minimal cylinders in terms of pairs of two closed plane curves which have the same signed area. Moreover, as a byproduct of the construction, spacelike CMC cylinders can also be obtained.