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

2012/10/27 by Dorfmeister, Josef F., Inoguchi, Jun-ichi, Kobayashi, Shimpei
#58D10 #Differential Geometry (math.DG) #FOS: Mathematics #Primary 53A10 #Secondary 53C42

paper · doi:10.48550/arxiv.1210.7300

Abstract

We characterize constant mean curvature surfaces in the three-dimensional Heisenberg group by a family of flat connections on the trivial bundle \D × \GL over a simply connected domain \mathbbD 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.

Related

• Minimal surfaces with non-trivial topology in the three-dimensional Heisenberg group
• Timelike Minimal Surfaces via Loop Groups
• Minimal cylinders in the three-dimensional Heisenberg group
• The Gauss map of Minimal graphs in the Heisenberg group
• Minimal Lagrangian surfaces in the two dimensional complex quadric via the loop group method