The Gauss map of minimal surfaces in the Heisenberg group

Details The Gauss map of minimal surfaces in the Heisenberg group The Gauss map of minimal surfaces in the Heisenberg group

2006-06-13

arxiv.org/abs/math/0606299v2

Int. Math. Res. Not. IMRN 2011 (2011), no. 3, 674-695

Mathematics

Differential Geometry

17 pages, redaction improved.

Scientific paper

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.

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