Faber–Krahn for the sub-Laplacian on the Heisenberg group

Determine whether balls (with respect to the sub-Riemannian structure) minimize the first Dirichlet eigenvalue of the Heisenberg sub-Laplacian −∑_{n=1}^{2N} Z_n^2 among open sets in ℍ^N of fixed Lebesgue measure.

Background

While Faber–Krahn-type results are classical in Euclidean settings, the corresponding question for sub-Riemannian operators on the Heisenberg group is substantially more delicate and depends on the geometry of horizontal distributions.

The paper explicitly notes that no analogue of the Rayleigh–Faber–Krahn theorem is known for the Heisenberg sub-Laplacian, underscoring a fundamental gap in the spectral geometry of sub-Riemannian spaces.