Fuglede's theorem in the Heisenberg group

Abstract

We prove a Fuglede-type quantitative isoperimetric inequality for the Pansu sphere, the conjectured isoperimetric set in the sub-Riemannian Heisenberg group $\mathbb{H}^1$. We show that, among suitable volume-preserving normal perturbations of the Pansu sphere that are small in a weighted $C^2$-norm, the difference between the sub-Riemannian perimeters of the perturbed and unperturbed regions is bounded below by a positive constant times the square of their $L^1$-distance. The proof is based on a Riemannian approximation argument with uniform estimates.