On Courant and Pleijel theorems for sub-Riemannian Laplacians

We are interested in the number of nodal domains of eigenfunctions of sub-Laplacians on sub-Riemannian manifolds. Specifically, we investigate the validity of Pleijel’s theorem, which states that the number of nodal domains of an eigenfunction corresponding to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> -th eigenvalue is strictly (and uniformly, in a certain sense) smaller than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> for large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> . We first reduce this question from the case of general sub-Riemannian manifolds to that of nilpotent groups. Secondly, we analyze in detail the case where the nilpotent group is a Heisenberg group times a Euclidean space. Along the way we improve known bounds on the optimal constants in the Faber–Krahn and isoperimetric inequalities on these groups. This is an announcement and the proofs will be given in a future paper (see also in ArXiv). This announcement is a modification with inclusion of new results of the more detailed announcement published in [10].