The Gauss map of minimal surfaces in the Heisenberg group

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.

17 pages, redaction improved.

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