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

TL;DR

This paper investigates the properties of the Gauss map for hypersurfaces in 2-step nilpotent Lie groups, deriving formulas for the Laplacian and conditions for harmonicity, especially in the Heisenberg group.

Contribution

It provides a general expression for the Laplacian of the Gauss map in 2-step nilpotent Lie groups and characterizes harmonic Gauss maps for CMC hypersurfaces in the Heisenberg group.

Findings

Derived Laplacian expression for the Gauss map

Established conditions for harmonic Gauss maps in the Heisenberg group

Classified CMC surfaces with harmonic Gauss maps as cylinders for m=1

Abstract

In this paper we consider smooth oriented hypersurfaces in 2-step nilpotent Lie groups with a left invariant metric and 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 derive necessary and sufficient conditions for the Gauss map to be harmonic and prove that for m=1 all CMC-surfaces with the harmonic Gauss map are "cylinders".