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

Abstract : 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 $\theta\mapsto \lambda_k(\theta)$, where $\lambda_k(\theta)$ is the $k$-th eigenvalue of the sub-Laplacian $\Delta_{b,\theta}$, and obtain necessary and sufficient conditions for a pseudo-Hermitian structure to be critical.
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Contributor : Ahmad El Soufi Connect in order to contact the contributor
Submitted on : Thursday, October 1, 2015 - 6:19:30 PM
Last modification on : Tuesday, January 11, 2022 - 5:56:09 PM
Long-term archiving on: : Saturday, January 2, 2016 - 11:27:56 AM

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Amine Aribi, Sorin Dragomir, Ahmad El Soufi. Eigenvalues of the sub-Laplacian and deformations of contact structures on a compact CR manifold: Eigenvalues of the sub-Laplacian. Differential Geometry and its Applications, Elsevier, 2015, 39, pp.113--128. ⟨10.1016/j.difgeo.2015.01.005⟩. ⟨hal-01208082⟩

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