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

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 $\theta \to \lambda_k(\theta)$, 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

The first positive eigenvalue of the sub-Laplacian on CR spheres

“…where V (g) denotes the Riemannian volume of the sphere with respect to g. Again, the equality holds in (2) if and only if g is isometric to g 0 . The aim of the present paper is to establish a version of the estimate (2) for the first positive eigenvalue of the sub-Laplacian on the CR sphere S 2n+1 ⊂ C n+1 .…”

“…This result can be seen as a contribution to the program aiming to recovering the main results of spectral geometry, established for the eigenvalues of the Laplace-Beltrami operator on a compact Riemannian manifold, in the realm of CR and pseudo-Hermitian geometry. This program has motivated a lot of research in recent years and we can find significant contributions in [1,2,3,4,6,5,7,8,9,10,13,14,17,16,18,19,20,21,22].…”

Second Variation Formula and Stability of Exponentially Subelliptic Harmonic Maps

“…For an arbitrary e.s.h. map φ we conjecture that J φ b , J φ b,exp and J φ b, exp have discrete spectra on domains ⊂ M enjoying appropriate 5 properties. This would of course lead to further developments of stability theory for e.s.h.…”

“…The continuity of the eigenvalues λ k (θ ) as functions of the contact form θ was established in [4]. The effect of deformations θ = e u θ , u ∈ C ∞ (M), on the eigenvalues λ k (θ ) was studied in [5]. For an arbitrary e.s.h.…”

“…(∂/∂ y A ) q = w A . Then [as b is a positive operator] 5 For instance, discreteness of the spectrum of the operator J φ b associated to a subelliptic harmonic map φ is established (cf. [11]) for a class of CR structures arising as orbit spaces M 3 of null Killing vector fields on a space-time (Gödel's universe in [11]), on a domain ⊂ M 3 supporting a form of Poincaré's inequality and a form of Kondrakov compactness involving L 2 ( , φ −1 T N).…”