The Gauss Map of Minimal Surfaces in the Heisenberg Group

Benoit Daniel (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

We study the Gauss map of minimal surfaces in the Heisenberg group 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 H(2). Conversely, any nowhere antiholomorphic harmonic map into 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.

Domaines: Physique mathématique [math-ph]

Benoit Daniel. The Gauss Map of Minimal Surfaces in the Heisenberg Group. International Mathematics Research Notices, 2011, ? (3), pp.674--695. ⟨10.1093/imrn/rnq092⟩. ⟨hal-00693002⟩