## Completeness, Reinhardt domains and the method of complex geodesics in the theory of invariant functions

### Volume 388 / 2000

#### Abstract

Our work is divided into five chapters. In Chapter I we introduce necessary notions and we present the most important facts that we shall use. We also present our main results. Chapter I covers the following topics: \vskip 4pt $\bullet$ holomorphically contractible families of functions and pseudometrics, their basic properties, product property, Lempert Theorem, notion of geodesic, problem of finding effective formulas for invariant functions and pseudometrics and geodesics, completeness with respect to holomorphically contractible distances, its application in the study of the relation between norm balls and Carathéodory balls; $\bullet$ pluricomplex Green function with a logarithmic pole as an example of a holomorphically contractible family of functions, problem of its symmetry, pluricomplex Green function with many poles as a natural generalization of the Green function with one pole; $\bullet$ Bergman distance, Bergman completeness. \vskip4pt Chapter II is devoted to the problem of completeness with respect to Carathéodory, Kobayashi and Bergman distances in a class of pseudoconvex Reinhardt domains. First we recall well known geometric properties of pseudoconvex Reinhardt domains (Section~2.1). In Section 2.2 we deal with properties of real convex cones, objects closely related to pseudoconvex Reinhardt domains. Section~2.3 is devoted to the study of algebraic mappings, especially those inducing proper and biholomorphic mappings of $\Bbb C_*^n$ (Theorem~2.3.1). A special role in our study will be played by quasi-elementary Reinhardt domains (Section 2.4). Before we study completeness we give a precise description of hyperbolic (in different sense) pseudoconvex Reinhardt domains (Theorem 2.5.1). The solution of the problem which hyperbolic pseudoconvex Reinhardt domains are Kobayashi (respectively, Carathéodory) complete, is given in Theorem 2.6.5 (respectively, Theorem 2.6.6). Additionally, the problem when the Carathéodory distance tends to infinity as one variable is fixed and the other tends to a boundary point not lying on axis in bounded pseudoconvex Reinhardt domains is discussed (Theorem 2.6.1, Corollary~2.6.2, and Example 2.6.4). In contrast to the Carathéodory and Kobayashi distances no characterization of Bergman completeness is known. Nevertheless, it is known in dimension $2$ (Corollary 2.7.4). Some partial results are given in Proposition 2.7.2 (a sufficient condition for not being Bergman complete) and Theorem 2.7.3 (a sufficient condition for Bergman completeness). A relation between good boundary behavior of the Green function and Bergman completeness in the class of bounded pseudoconvex Reinhardt domains (Lemma 2.8.2 and Proposition 2.8.5) and in planar domains (Corollary 2.8.8) is considered. In Chapter III we find formulas for holomorphically contractible functions and pseudometrics in the class of elementary Reinhardt domains (Sections 3.1–3.5) and for the pluricomplex Green function of the unit ball with two poles (with equal weights) (Section~3.6). First we recall known formulas (Theorem 3.1). Then we present formulas for elementary Reinhardt domains not contained in $\Bbb C_*^n$ (Theorem 3.1.1). The proof of the theorem is contained in Sections 3.2–3.4. For elementary Reinhardt domains lying in $\Bbb C_*^n$ the proof of the formulas (Theorem 3.5.1) is much simpler. Theorem 3.6.1 gives a formula for the pluricomplex Green function of the unit ball with two poles of equal weights. The key role in the proof of the formula is played by Theorem 3.6.2 showing how the pluricomplex Green function with many poles behaves under proper holomorphic mappings. In Chapter IV we deal with symmetry of the Green function. First we entirely solve the problem in the class of complex ellipsoids (Theorem 4.1.1). In Section 4.2 some kind of “infinitesimal” symmetry in the class of bounded hyperconvex domains is described (Corollary 4.2.4). This property is a consequence of regularity properties of the Azukawa pseudometric (Theorems 4.2.1 and 4.2.2, and Corollary 4.2.3). The results on regularity properties of the Azukawa pseudometric cannot be extended to the class of bounded pseudoconvex domains (Example 4.2.10). In Section 4.3 we discuss the problem of nonsymmetry of the Green function in pseudoconvex complete Reinhardt domains whose boundary contains some “exponential line”. It turns out that in such domains the Green function is extremely nonsymmetric (Propositions 4.3.1 and 4.3.2, and Remark~4.3.3). In Chapter V we consider the problem which Carathéodory balls are simultaneously norm balls in the class of convex ellipsoids. The ideas used in this chapter have been used lately in the study of the same problem for a wider class of domains. Most of the properties that we use may be found in the following books: \cite{Kob~70}, \cite{Kli~91}, \cite{Jar-Pfl~93}, and \cite{Kob~98}. If some result that we use is not quoted explicitly it may be found in one of these books. Some of the results contained in the work may be found in the following papers: {\ifx \pol\ver \cite{Zwo~96},\fi} \cite{Edi-Zwo~98}, \cite{Pfl-Zwo~98}, \cite{Zwo~97}, \cite{Zwo~98a}, \cite{Zwo~98b}, \cite{Zwo~98c}, and \cite{Zwo~99}. \vskip 4pt plus 2pt While writing the paper the author was a fellow of the Alexander von Humboldt Foundation. The author was also supported by the KBN grant No. 2 P03A 017 14. The author would like to thank Professors M. Jarnicki and P. Pflug for their remarks on earlier versions of the work (especially for pointing out some errors), which essentially improved the final version, and for stimulating discussions concerning the subject. The author is also grateful to Z. B/locki and A. Edigarian for fruitful discussions.