Constant mean curvature hypersurfaces in a Lie group with a bi-invariant metric

Abstract

Given an orientable hypersurface M of a Lie group 𝔾 with a bi-invariant metric we consider the map N : M → 𝕊 n that translates the normal vector field of M to the identity, which is a natural extension of the usual Gauss map of hypersurfaces in Euclidean spaces; it is proved that the Laplacian of N satisfies a formula similar to that satisfied by the usual Gauss map. One may then conclude that M has constant mean curvature (cmc) if and only if N is harmonic; some other aplications to cmc hypersurfaces of 𝔾 are also obtained.