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

TL;DR

This paper classifies umbilic hypersurfaces with constant sigma-k curvature in the Heisenberg group, showing they are rotationally invariant and identifying Pansu spheres as unique positive curvature examples.

Contribution

It provides a classification of umbilic hypersurfaces with constant sigma-k curvature in the Heisenberg group, highlighting the uniqueness of Pansu spheres.

Findings

Closed umbilic hypersurfaces are rotationally invariant up to translation.

Pansu spheres are the only positive constant sigma-k curvature spheres.

Such hypersurfaces are characterized by their invariance properties.

Abstract

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.