Eigenvalues of the sub-Laplacian and deformations of contact structures on a compact CR manifold

Given a compact strictly pseudoconvex CR manifold $M$, we study the differentiability of the eigenvalues of the sub-Laplacian $\Delta_{b, theta}$ associated with a compatible contact form (i.e. a pseudo-Hermitian structure) $\theta$ on $M$, under conformal deformations of $\theta$. As a first application, we show that the property of having only simple eigenvalues is generic with respect to $\theta$, i.e. the set of structures $\theta$ such that all the eigenvalues of $\Delta_{b,\theta}$ are simple, is residual (and hence dense) in the set of all compatible positively oriented contact forms on $M$. In the last part of the paper, we introduce a natural notion of critical pseudo-Hermitian structure of the functional $\theta \to \lambda_k(\theta)$, where $\lambda_k(\theta)$ is the $k$-th eigenvalue of the subì-Laplacian $\Delta_{b, \theta}$ , and obtain necessary and sufficient conditions for a pseudo-Hermitian structure to be critical.