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

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