A loop group method for minimal surfaces in the three-dimensional Heisenberg group

We characterize constant mean curvature surfaces in the three-dimensional Heisenberg group by a family of flat connections on the trivial bundle $\D \times \GL$overasimplyconnecteddomain$$\mathbb{D}$$ in the complex plane. In particular for minimal surfaces, we give an immersion formula, the so-called Sym-formula, and a generalized Weierstrass type representation via the loop group method.