##
**Proper holomorphic mappings: A survey.**
*(English)*
Zbl 0778.32008

Several complex variables, Proc. Mittag-Leffler Inst., Stockholm/Swed. 1987-88, Math. Notes 38, 297-363 (1993).

[For the entire collection see Zbl 0759.00008.]

This is a very comprehensive survey on the subject written by one of the best specialist who had given deep contributions in the last 7-8 years. In my opinion, the paper represents an absolutely necessary tool either for the specialist (he will find an exhaustive exposition of the main results), either for the non-specialist who needs general informations and will get a large vision of the development of the subject from its beginning up to day. Let us mention that the bibliography contains 177 titles!

Since the literature on this subject is growing very rapidly, the author had to omit certain topics in order not to make the paper excessively long. The selection reflects the personal choices of the author. Let us follow him:

From the introduction: “In section 1 we survey the results on the regularity at the boundary of proper holomorphic mappings between smoothly bounded domains in \(\mathbb{C}^ n\). This has been a very active area of research ever since Fefferman proved in 1974 that biholomorphic mappings between smooth bounded strongly pseudoconvex domains in \(\mathbb{C}^ n\) extend smoothly to the boundary. We first survey the results obtained by the method of the Bergman kernel and the related \(\overline\partial\)-Neumann problem. Most of these results have been covered in a survey by Bedford. There are some new results concerning the condition \(R\), the most interesting ones due to Boas and Staube, as well as new regularity results for locally proper mappings, due to Bell and Catlin. We also present an elementary approach to the regularity problem for mappings of strongly pseudoconvex domains, due to Pincuk and Hasanov and Forstnic, that reduces the problem to the \({\mathcal C}^ \infty\) version of the edge-of-the-wedge theorem. There is a new regularity result of Pincuk and Tsyganov for continuous \(CR\) mappings between strongly pseudoconvex hypersurfaces, and an optimal regularity result due to Hasanov.

In section 2 we consider the mappings between bounded domains in \(\mathbb{C}^ n\) with real-analytic boundaries; the main problem is to show that such mappings extend biholomorphically across the boundary. In one variable this is the classical Schwarz reflection principle. In several variables, this phenomenon was first discovered by Lewy and Pincuk in the case of strongly pseudoconvex boundaries. Very interesting and far reaching generalizations were obtained in recent years by several authors, and the research in this field is still very intensive. Most notably, the problem has been solved on pseudoconvex domains in \(\mathbb{C}^ n\) by Baouendi and Rothschild and Diederich and Fornaess.

In section 3 we collect results on mappings between some special classes of domains like the ball, the circular domains, the bounded symmetric domains, the Reinhardt domains, the ellipsoids, and the generalized ellipsoids. Besides the well-known results we present some of the recent developments. Perhaps the most interesting new result is the complete solution of the equivalence problem for Reinhardt domains by Shimizu.

In section 4 we present rather recent results on existence of proper holomorphic mappings of domains into special higher-dimensional domains like the ball and the polydisc. These new constructions of proper mappings, due mainly to Løw, Forstneric and Stensønes, were motivated by the construction of inner functions in early 1980s. The initial construction has been improved substantially by Stensønes.

Section 5 contains regularity results for mappings into higher- dimensional domains. Results in this field are still rather fragmentary in spite of some recent progress. Most of the results of sections 4 and 5 have been obtained since 1985. The reader should compare the surveys by Bedford from 1984 and by Cima and Suffridge from 1987.

In section 6 we present results on the classification of proper holomorphic mappings between balls. Although there has been a lot of progress in this direction recently, mainly due to D’Angelo, we do not have a systematic theory yet. Because of its intrinsic beauty and its interesting connections with other areas of mathematics, this problem would deserve more attention”.

This is a very comprehensive survey on the subject written by one of the best specialist who had given deep contributions in the last 7-8 years. In my opinion, the paper represents an absolutely necessary tool either for the specialist (he will find an exhaustive exposition of the main results), either for the non-specialist who needs general informations and will get a large vision of the development of the subject from its beginning up to day. Let us mention that the bibliography contains 177 titles!

Since the literature on this subject is growing very rapidly, the author had to omit certain topics in order not to make the paper excessively long. The selection reflects the personal choices of the author. Let us follow him:

From the introduction: “In section 1 we survey the results on the regularity at the boundary of proper holomorphic mappings between smoothly bounded domains in \(\mathbb{C}^ n\). This has been a very active area of research ever since Fefferman proved in 1974 that biholomorphic mappings between smooth bounded strongly pseudoconvex domains in \(\mathbb{C}^ n\) extend smoothly to the boundary. We first survey the results obtained by the method of the Bergman kernel and the related \(\overline\partial\)-Neumann problem. Most of these results have been covered in a survey by Bedford. There are some new results concerning the condition \(R\), the most interesting ones due to Boas and Staube, as well as new regularity results for locally proper mappings, due to Bell and Catlin. We also present an elementary approach to the regularity problem for mappings of strongly pseudoconvex domains, due to Pincuk and Hasanov and Forstnic, that reduces the problem to the \({\mathcal C}^ \infty\) version of the edge-of-the-wedge theorem. There is a new regularity result of Pincuk and Tsyganov for continuous \(CR\) mappings between strongly pseudoconvex hypersurfaces, and an optimal regularity result due to Hasanov.

In section 2 we consider the mappings between bounded domains in \(\mathbb{C}^ n\) with real-analytic boundaries; the main problem is to show that such mappings extend biholomorphically across the boundary. In one variable this is the classical Schwarz reflection principle. In several variables, this phenomenon was first discovered by Lewy and Pincuk in the case of strongly pseudoconvex boundaries. Very interesting and far reaching generalizations were obtained in recent years by several authors, and the research in this field is still very intensive. Most notably, the problem has been solved on pseudoconvex domains in \(\mathbb{C}^ n\) by Baouendi and Rothschild and Diederich and Fornaess.

In section 3 we collect results on mappings between some special classes of domains like the ball, the circular domains, the bounded symmetric domains, the Reinhardt domains, the ellipsoids, and the generalized ellipsoids. Besides the well-known results we present some of the recent developments. Perhaps the most interesting new result is the complete solution of the equivalence problem for Reinhardt domains by Shimizu.

In section 4 we present rather recent results on existence of proper holomorphic mappings of domains into special higher-dimensional domains like the ball and the polydisc. These new constructions of proper mappings, due mainly to Løw, Forstneric and Stensønes, were motivated by the construction of inner functions in early 1980s. The initial construction has been improved substantially by Stensønes.

Section 5 contains regularity results for mappings into higher- dimensional domains. Results in this field are still rather fragmentary in spite of some recent progress. Most of the results of sections 4 and 5 have been obtained since 1985. The reader should compare the surveys by Bedford from 1984 and by Cima and Suffridge from 1987.

In section 6 we present results on the classification of proper holomorphic mappings between balls. Although there has been a lot of progress in this direction recently, mainly due to D’Angelo, we do not have a systematic theory yet. Because of its intrinsic beauty and its interesting connections with other areas of mathematics, this problem would deserve more attention”.

Reviewer: I.P.Ramadanov (Caen)

### MSC:

32H35 | Proper holomorphic mappings, finiteness theorems |

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