# Minimal surfaces with non-trivial topology in the three-dimensional Heisenberg group

## TL;DR

This paper explores the construction and classification of symmetric minimal surfaces with non-trivial topology in the three-dimensional Heisenberg group using advanced mathematical methods.

## Contribution

It introduces a method to construct and classify minimal surfaces with complex topology in the Heisenberg group using the loop group approach.

## Key findings

- Constructed explicit examples of minimal surfaces with non-trivial topology.
- Classified equivariant minimal surfaces under one-parameter subgroup symmetries.
- Demonstrated the effectiveness of the loop group method in this geometric context.

## Abstract

We study symmetric minimal surfaces in the three-dimensional Heisenberg group $\mathrm{Nil}_3$ using the generalized Weierstrass type representation, the so-called loop group method. In particular, we will discuss how to construct minimal surfaces in $\mathrm{Nil}_3$ with non-trivial topology. Moreover, we will classify equivariant minimal surfaces given by one-parameter subgroups of the isometry group $\mathrm{Iso}_{\circ}(\mathrm{Nil}_3)$ of $\mathrm{Nil}_3$.