##
**Real submanifolds in complex space and their mappings.**
*(English)*
Zbl 0944.32040

Princeton Mathematical Series 47. Princeton, NJ: Princeton University Press (ISBN 0-691-00498-6/hbk). xii, 404 p. (1999).

This is a very nice book which surveys many of the recent studies on real submanifolds in complex spaces and their mappings. The authors themselves are very well-known active researchers, who have been making substantial contributions to the topics covered here.

The book starts with the basic materials concerning the CR structures and correponding CR functions on them. Then they discuss the extension of CR functions to a certain side, using the highly powerful methods of analytic disks and Baouendi-Treves approximation. Then the authors give a very detailed study for the two side extension for \(R\) mappings, including the proof of several important theorems in the subject.

In the last part, the authors study the rigidity phenomenon for mappings between algebraic submanifolds, initiated in the foundational work of PoincarĂ© and Webster. Over all, this is an extremelly well-writen book with complete and careful proofs to many fundamental facts in the field which are difficult to find in the references.

The authors also give a precise discussion on the history of the research. The book should serve as a basic reference book in the subject of several complex variables.

The book starts with the basic materials concerning the CR structures and correponding CR functions on them. Then they discuss the extension of CR functions to a certain side, using the highly powerful methods of analytic disks and Baouendi-Treves approximation. Then the authors give a very detailed study for the two side extension for \(R\) mappings, including the proof of several important theorems in the subject.

In the last part, the authors study the rigidity phenomenon for mappings between algebraic submanifolds, initiated in the foundational work of PoincarĂ© and Webster. Over all, this is an extremelly well-writen book with complete and careful proofs to many fundamental facts in the field which are difficult to find in the references.

The authors also give a precise discussion on the history of the research. The book should serve as a basic reference book in the subject of several complex variables.

Reviewer: Xiaojun Huang (New Brunswick)

### MSC:

32V40 | Real submanifolds in complex manifolds |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

32V10 | CR functions |

32V25 | Extension of functions and other analytic objects from CR manifolds |

32V05 | CR structures, CR operators, and generalizations |