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The space of Cauchy-Riemann structures on 3-D compact contact manifolds. (English) Zbl 1244.32018

The authors study the action of the group of contact diffeomorphisms on \(CR\) deformations of compact \(3\)-dimensional strongly pseudoconvex \(CR\) manifolds. Starting with an embeddable \((M,\bar\partial_b)\), they prove that there is an open neighborhood in the space of \(CR\) structures in which all deformations tensors have the normal form \(F^*_{\Psi{X}}\phi=i\bar\partial_bY+\psi\), where \(i\bar\partial_bY\) represents a Kuranishi wiggle, i.e. a deformation arising from deforming the embedding of \((M,\bar\partial_b)\) in its ambient surface through a diffeomorphism; \(\psi\) represents the obstruction to embeddability and \(\Psi{X}\) is a contact diffeomorphism corresponding to a contact vector field \(X\). Of course, using the stability theorem, the contact structure is kept fixed in the deformation. The map \(\phi@>>> (X_{\phi},Y_{\phi},\psi_{\phi})\) is shown to be continuous in the \(\Gamma^s\) norms, for \(s\geq{6}\), where the \(\Gamma^s\) are the Folland-Stein anisotropic Sobolev spaces.
The second main result concerns regularity. The linearization of the normalizing map at \(\phi=0\) is \(\bar\partial_b(X-iY)+\psi\), and, using homotopy operators, it is shown that, when \(\phi\in\Gamma^s\), then \(X,Y\in\Gamma^{1+s}\), \(\psi\in\Gamma^s\). These regularity results also fix some gaps of [J. S. Bland, Acta Math. 172, No. 1, 1–49 (1994; Zbl 0814.32002)].

MSC:

32V15 CR manifolds as boundaries of domains
32V30 Embeddings of CR manifolds
32V05 CR structures, CR operators, and generalizations
53D10 Contact manifolds (general theory)

Citations:

Zbl 0814.32002
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