Umbilic hypersurfaces of constant sigma-k curvature in the Heisenberg group

Jih-Hsin Cheng, Hung-Lin Chiu, Jenn-Fang Hwang, Paul Yang

Published 2015-12-18Version 1

We study immersed, connected, umbilic hypersurfaces in the Heisenberg group $H_{n}$ with $n$ $\geq $ $2.$ We show that such a hypersurface, if closed, must be rotationally invariant up to a Heisenberg translation. Moreover, we prove that, among others, Pansu spheres are the only such spheres with positive constant sigma-k curvature up to Heisenberg translations.