zbMATH — the first resource for mathematics

Transversal Lie group actions on abstract CR manifolds. (English) Zbl 0673.32021
It is shown that if an abstract CR manifold M has a local transversal Lie group action, then M is locally embeddable as a generic submanifold of \({\mathbb{C}}^ n\). Under the additional assumption that the infinitesimal action of the group forms a subbundle of TM, it is also shown that the action has a local holomorphic extension to \({\mathbb{C}}^ n\) in this embedding. Furthermore, it is shown that there is a unique embedding with this property, up to biholomorphism. Finally, if the action is strictly transversal to the CR structure, it is proved that any CR function can be locally decomposed as a finite sum of boundary values of holomorphic functions in open wedges with edge M.
Reviewer: M.S.Baouendi

32V40 Real submanifolds in complex manifolds
32M05 Complex Lie groups, group actions on complex spaces
57S20 Noncompact Lie groups of transformations
57S15 Compact Lie groups of differentiable transformations
Full Text: DOI EuDML
[1] Baouendi, M.S., Chang, C.H., Treves, F.: Microlocal hypo-analyticity and extension of CR functions. J. Differ. Geom.18, 331-391 (1983) · Zbl 0575.32019
[2] Baouendi, M.S., Rothschild, L.P.: Analytic approximation for homogeneous solutions of invariant differential operators on Lie groups. Asterisque131, 189-199 (1985) · Zbl 0597.58048
[3] Baouendi, M.S., rothschild, L.P.: Normal forms for generic manifolds and holomorphic extension of CR functions. J. Differ. Geom.25, 431-467 (1987) · Zbl 0629.32016
[4] Baouendi, M.S., Rothschild, L.P.: Embeddability of abstract CR structures and integrability of related systems. Ann. Inst. Fourier37, 131-141 (1987) · Zbl 0619.58001
[5] Baouendi, M.S., Rothschild, L.P.: Extension of holomorphic functions in generic wedges and their wave front sets. Commun. Part. Differ. Equations13, 1441-1466 (1988) · Zbl 0664.32010 · doi:10.1080/03605308808820583
[6] Baouendi, M.S., Rothschild, L.P., Treves, F.: CR structures with group action and extendability of CR functions. Invent. Math.82, 359-396 (1985) · Zbl 0598.32019 · doi:10.1007/BF01388808
[7] Jacobowitz, H.: The canonical bundle and realizable CR hypersurfaces. (preprint) · Zbl 0583.32050
[8] Newlander, A., Nirenberg, L.: Complex coordinates in almost complex manifolds. Ann. Math.65, 391-404 (1957) · Zbl 0079.16102 · doi:10.2307/1970051
[9] Nirenberg, L.: A Complex Frobenius Theorem. Seminar on analytic functionsI, pp. 172-189. Princeton (1957)
[10] Nomizu, K.: Lie groups and differential geometry. Math. Soc. Japan (1956) · Zbl 0071.15402
[11] Sj?strand, J.: Singularit?s analytiques microlocales. Ast?risque95, 1-166 (1982)
[12] Tanaka, N.: On the pseudoconformal geometry of hypersurfaces of the space ofn complex variables. J. Math. Soc. Japan14, 397-429 (1962) · Zbl 0113.06303 · doi:10.2969/jmsj/01440397
[13] Trepreau, J.-M.: Une fonction CR non d?composable en somme de valeurs au bord de fonctions holomorphes, (unpublished)
[14] Varadarajan, V.S.: Lie groups, Lie algebras, and their representations. Englewood Cliffs, NJ: Prentice-Hall 1974 · Zbl 0371.22001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.