The Classification of Rotationally symmetric hypersurfaces in the Heisenberg groups $H_n$

In this paper, we show the fundamental theorems for rotationally symmetric hypersurfaces, and thus, together with the earlier results in [3] and [4], provide a complete classification of umbilic hypersurfaces in the Heisenberg groups $H_n$. In addition, we give a complete description of generating curves for rotationally symmetric hypersurfaces with constant $p$-mean curvature $H=c$ (including $H=0$) in the Heisenberg group $H_n$. We also establish the validity of Alexandrov’s theorem for rotationally symmetric hypersurfaces in $H_n$.