Embedding compact three-dimensional CR manifolds of finite type in \({\mathbb{C}}^ n\). (English) Zbl 0668.32019

Christ, Michael

Ann. Math. (2) 129, No. 1, 195-213 (1989).

The author proves that any smooth, compact, three dimensional CR manifold, X, which is pseudoconvex and of finite type, and on which \(\overline{\partial}_ b\) has closed range in \(L^ 2\) admits a CR embedding into some \({\mathbb{C}}^ n\). The author refers to the literature for the corresponding result in case X is strongly pseudoconvex, in this paper he is able to replace the hypothesis of strong pseudoconvexity with that of pseudoconvexity plus finite type.
In order to obtain CR functions which separate points, the author constructs a one parameter family of CR functions all equal to one at a given point on X and tending to O elsewhere. He does this first for certain model hypersurfaces in \({\mathbb{C}}^ 2\) of the form \(M=\{(Z_ 1,Z_ 2):\) \(Im(Z_ 2)=P(Z_ 1)\}\), for a real valued homogeneous polynomial P. Near a point on X he shows how to choose coordinates so as to exhibit X as a perturbation of such a model. He then derives suitable CR functions on X via an approximation argument, the main tool being the \(L^ p\) Sobolev and Hölder regularity estimates for \(\overline{\partial}_ b\) developed by C. L. Fefferman and J. J. Kohn [Adv. Math. 69, 223-303 (1988; Zbl 0649.35068)] and by himself [J. Am. Math. Soc. 1, No.3, 587-646 (1988)].

Reviewer: G.Harris

MSC:

32V40	Real submanifolds in complex manifolds
35N15	\(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
32W05	\(\overline\partial\) and \(\overline\partial\)-Neumann operators

Keywords:

CR manifold; pseudoconvex; finite type; CR embedding; CR functions

Citations:

Zbl 0649.35068