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 Δb,θΔb,θ associated with a compatible contact form (i.e. a pseudo-Hermitian structure) θ on M, under conformal deformations of θ. As a first application, we show that the property of having only simple eigenvalues is generic with respect to θ, i.e. the set of structures θ   such that all the eigenvalues of Δb,θΔb,θ 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 θ↦λk(θ)θ↦λk(θ), where λk(θ)λk(θ) is the k  -th eigenvalue of the sub-Laplacian Δb,θΔb,θ, and obtain necessary and sufficient conditions for a pseudo-Hermitian structure to be critical.