CMC spheres in the Heisenberg group

TL;DR

This paper investigates constant mean curvature spheres in the Heisenberg group, providing evidence for their role as isoperimetric sets and analyzing their behavior in the sub-Riemannian limit.

Contribution

It offers new results supporting the conjecture that CMC spheres are isoperimetric in the Heisenberg group and explores their properties in the sub-Riemannian limit.

Findings

Support for the isoperimetric conjecture in H^1

Analysis of CMC spheres in the sub-Riemannian limit

New geometric properties of CMC spheres in H^1

Abstract

We study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group $H^1$. These spheres are conjectured to be the isoperimetric sets of $H^1$. We prove several results supporting this conjecture. We also focus our attention on the sub-Riemannian limit.