The Classification of Rotationally symmetric hypersurfaces in the Heisenberg groups H_n

2025/09/05 by Chiu, Hung-Lin, Lai, Sin-Hua, Liu, Hsiao-Fan
#34A26 #53A10 #53C22 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.04972

Abstract

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.

Citations

• A characterization of constant p-mean curvature surfaces in the Heisenberg group H₁
• Umbilic hypersurfaces of constant sigma-k curvature in the Heisenberg group
• Umbilicity and characterization of Pansu spheres in the Heisenberg group
• Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group
• The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations
• Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group Hⁿ

Related

• Umbilic hypersurfaces of constant sigma-k curvature in the Heisenberg group
• Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group Hⁿ
• Umbilicity and characterization of Pansu spheres in the Heisenberg group
• A characterization of horizontally totally geodesic hypersurfaces in Heisenberg groups
• The fundamental theorem of curves and classifications in the Heisenberg groups