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

Abstract

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.

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 added

References in corpus (2)

• Minimal translation surfaces in the Heisenberg group Nil3
• Surfaces in three-dimensional Lie groups in terms of spinors

Cited by in corpus (3)

• A loop group method for affine harmonic maps into Lie groups
• Willmore-like functionals for surfaces in 3-dimensional Thurston geometries
• Sym-Bobenko formula for minimal surfaces in Heisenberg space