CR manifolds and the tangential Cauchy Riemann complex. (English) Zbl 0760.32001

Boggess, Albert

Studies in Advanced Mathematics. Boca Raton, FL: CRC Press. xvii, 364 p. (1991).

The purpose of the present book is to define CR manifolds and the associated tangential Cauchy-Riemann complex and to discuss some of their properties. The material is presented in four parts. In the first part, the author provides the prerequisites for reading the rest of the book. Here we find several concepts as: functions, vector fields, differential forms on both Euclidean spaces and manifolds. There are given proofs for Stokes’ theorem and Frobenius theorem. It is also discussed the distribution theory as applied to partial differential equations and the distribution theory for differential forms, that is, the theory of currents. The second part is dealing with the general theory of CR manifolds. There are presented both abstract and imbedded CR manifolds. Then there are introduced and discussed the concepts of tangential Cauchy-Riemann complex and the CR function. Finally, there are studied the Levi form and the problem of imbeddability of CR manifolds. In the third part, it is discussed the local holomorphic extension of CR functions from an imbedded CR manifold. First, it is given an approximation theorem which states that CR functions can be locally approximated by entire functions. Then there are presented both the Hans Lewy’s CR extension theorem for hypersurfaces and the CR extension theorem for higher dimension, the latter being generalized to CR distributions. The last part of the book is concerned with global and local solvability of the tangential Cauchy-Riemann operator on a CR manifold. Here, first the calculus of kernels and then various fundamental solutions for \(d\) and \(\overline\partial\) are discussed. Then the Bochner’s global CR extension theorem is proved and Henkin’s kernels are introduced. Finally, it is given the Hans Lewy’s local nonsolvability example.
In the reviewer’s opinion the present book is an important contribution to the literature of several complex variables and partial differential equations. The audience includes both researchers and graduate students.

Reviewer: A.Bejancu (Iaşi)

MSC:

32-02	Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32V99	CR manifolds
32V05	CR structures, CR operators, and generalizations
35N15	\(\overline\partial\)-Neumann problems and formal complexes in context of PDEs

Keywords:

CR manifold; CR function; Levi form; tangential Cauchy-Riemann operator