Asian Journal of Mathematics

Volume 20 (2016)

Number 3

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

Pages: 409 – 448

DOI: https://dx.doi.org/10.4310/AJM.2016.v20.n3.a2

Authors

Josef F. Dorfmeister (Fakultät für Mathematik, Technische Universität München, Germany)

Jun-Ichi Inoguchi (Institute of Mathematics, Tsukuba University, Tsukuba, Japan)

Shimpei Kobayashi (Department of Mathematics, Hokkaido University, Sapporo, Japan)

Abstract

We characterize constant mean curvature surfaces in the three-dimensional Heisenberg group by a family of flat connections on the trivial bundle $\mathbb{D} \times \mathrm{GL}_2 \mathbb{C}$ 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. Our generalized Weierstrass type representation produces all simply-connected non-vertical minimal surfaces in the Heisenberg group.

Keywords

constant mean curvature, Heisenberg group, spinors, generalized Weierstrass type representation

2010 Mathematics Subject Classification

Primary 53A10, 58D10. Secondary 53C42.

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Published 12 July 2016