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 over a simply connected domain $\mathbb{D}$ 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.Comment: 40 pages, v2: The argument for branch points has been fixed and the references have been updated. v3: The argument for canonical examples has been fixed and the classification for homogeneous surfaces has been added v4: Some typos are fixed and Remark 5.4 (3) is adde