On Courant and Pleijel theorems for sub-Riemannian Laplacians

TL;DR

This paper examines the behavior of nodal domains of eigenfunctions of sub-Laplacians on sub-Riemannian manifolds, extending Pleijel's theorem to this setting and analyzing specific cases like Heisenberg groups.

Contribution

It reduces the general problem to nilpotent groups and provides detailed analysis for Heisenberg groups, improving bounds on related inequalities.

Findings

Pleijel's theorem holds for sub-Riemannian Laplacians in higher dimensions.

Reduced the problem to nilpotent groups, simplifying analysis.

Improved bounds on Faber-Krahn and isoperimetric constants for these groups.

Abstract

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.