A loop group method for minimal surfaces in the three-dimensional Heisenberg group (Q317505)

The three-dimensional Heisenberg group \(\mathrm{Nil}_3\) is a nilpotent Lie group equipped with a left-invariant metric, which is also a contact manifold with a unit Killing vector field, called the Reeb vector field.NEWLINENEWLINEIn this paper, the authors first characterize constant mean curvature surfaces in \(\mathrm{Nil}_3\) by a family of flat connections on the trivial \(\mathrm{GL}_2\mathbb C\) bundle over a simply connected domain in the complex plane. In particular, characterizing minimal surfaces nowhere tangent to the Reeb vector field by a family of flat connections on the \(\mathrm{SU}_{1,1}\) bundle, they give an immersion formula in terms of a family of \(\mathrm{SU}_{1,1}\)-valued frames.NEWLINENEWLINEFinally, together with explicit examples such as horizontal umbrellas, hyperbolic paraboloids, helicoids, and catenoids, they obtain a generalized Weierstrass-type representation for minimal surfaces in \(\mathrm{Nil}_3\) via a loop group method, which recovers minimal surfaces by a pair of holomorphic functions through the loop group decomposition.