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

Abstract

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 2 C over a simply connected domain 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.

On the Bernstein Problem in the Three-dimensional Heisenberg Group

“…Therefore we will sometimes omit the word "vertical". In Appendix B, we give a short review of facts and results of [5] which are used for the solution to the Bernstein problem via loop groups. From now on, we denote the coordinates of Nil 3 or L 3 by (x 1 , x 2 , x 3 ).…”

“…respectively. It is known that the vector of generating spinorsψ = (ψ 1 , ψ 2 ) satisfies the so-called "linear spinor system" [5]:…”

“…Every simply connected (nowhere vertical) minimal surface is obtained from an extended frame for a harmonic map into the hyperbolic two-space H 2 . In this paper we give a short proof of the solution to the Bernstein problem in Nil 3 by virtue of the generalized Weierstrass type representation established in [5]. The advantage of our approach is that we can give a direct relation between minimal graphs in Nil 3 and spacelike CMC surface with mean curvature H = 1/2 graphs in L 3 , Theorem 1.7.…”

“…and a nilpotent Lie group structure, see for example [5]. The Riemannian metric is invariant under the nilpotent Lie group structure and has a 4-dimensional isometry group.…”

“…Our proof is much more direct. In our previous work [5], we have established a generalized Weierstrass type representation for minimal surfaces in Nil 3 . Every simply connected (nowhere vertical) minimal surface is obtained from an extended frame for a harmonic map into the hyperbolic two-space H 2 .…”