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Chanillo, Sagun; Chiu, Hung-Lin; Yang, Paul

Embeddability for 3-dimensional Cauchy-Riemann manifolds and CR Yamabe invariants. (English) Zbl 1271.32040

Duke Math. J. 161, No. 15, 2909-2921 (2012).
It is well known that the question of embeddability of a hypersurface type CR structure has special character in dimension \(3\). In this article, the authors derive sufficient conditions for global embeddability of a \(3\)-dimensional CR manifold into some \(\mathbb C^n\) in terms of CR-covariant differential operators. The main role is played by the CR-Paneitz operator, which is of order \(4\). The main result is that non-negativity of the Paneitz operator together with positivity of the Webster-Tanaka curvature (or of the CR-Yamabe invariant) implies global embeddability.
The main ingredient in the proof is a Bochner formula for the Kohn-Laplacian, in which the pseudo-Hermitian torsion does not occur. Via this formula, the above assumptions are sufficient to derive a lower bound on non-zero eigenvalues of the Kohn-Laplacian. This in turn implies closedness of its range and then, via a theorem of Kohn, global embeddability. Finally, the authors prove a result on stability of an embeddable \(3\)-dimensional CR structure.
Reviewer: Andreas Cap (Wien)

Cited in 3 Reviews
Cited in 33 Documents

MSC:

32V30 Embeddings of CR manifolds
32V20 Analysis on CR manifolds
32V05 CR structures, CR operators, and generalizations

Keywords:

3-dimensional CR structure; global embeddability; CR invariant differential operator; CR Paeneitz operator; Kohn Laplacian
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References:

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