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  1. ; (, Journal de l’École polytechnique — Mathématiques)
    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, as soon as the dimension is strictly larger than 1 , the number of nodal domains of an eigenfunction corresponding to the k -th eigenvalue is strictly (and uniformly, in a certain sense) smaller than k for large  k . In the first part of this paper we reduce this question from the case of general sub-Riemannian manifolds to that of nilpotent groups. In the second part, 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. 
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