The Gauss map of minimal surfaces in the Heisenberg group

Benoît Daniel

The Gauss map of minimal surfaces in the Heisenberg group

Differential Geometry 2011-03-23 v2

Authors: Benoît Daniel

Abstract

We study the Gauss map of minimal surfaces in the Heisenberg group $\mathrm{Nil}_3$ endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane $\mathbb{H}^2$ . Conversely, any nowhere antiholomorphic harmonic map into $\mathbb{H}^2$ is the Gauss map of a nowhere vertical minimal surface. Finally, we study the image of the Gauss map of complete nowhere vertical minimal surfaces.

Keywords

gauss map minimal surfaces heisenberg group harmonic map hyperbolic plane

Cite

@article{arxiv.math/0606299,
  title  = {The Gauss map of minimal surfaces in the Heisenberg group},
  author = {Benoît Daniel},
  journal= {arXiv preprint arXiv:math/0606299},
  year   = {2011}
}

Comments

17 pages, redaction improved.

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