The Gauss Map of Hypersurfaces in 2-Step Nilpotent Lie Groups

Abstract

We consider smooth oriented hypersurfaces in 2-step nilpotent Lie groups with a left invariant metric. We derive an expression for the Laplacian of the Gauss map for such hypersurfaces in the general case and in some particular cases. In the case of CMC-hypersurface in the 2m+1-dimensional Heisenberg group we also give necessary and su cient conditions for the Gauss map to be harmonic and prove that for m = 1 all CMC-surfaces with the harmonic Gauss map are cylinders .