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

We study immersed, connected, umbilic hypersurfaces in the Heisenberg group \( H_{n}\) with n \(\ge \) 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.