Bergman completeness of unbounded hartogs Domains

Abstract

Abstract. Some results for the Bergman functions in unbounded domains are shown. In particular, a class of unbounded Hartogs domains, which are Bergman complete and Bergman exhaustive, is given.

$$L_h^2$$ L h 2 -Functions in Unbounded Balanced Domains

“…Remark 18. Examples that were studied in [15] show that it is a kind of the Lelong number which may be responsible for the fact that the balanced domain is Bergman complete. More precisely, the following description of Bergman complete balanced domains in C 2 may be correct: D = D h is Bergman complete iff ν(log, [v]) = 0 for any [v] ∈ P 1 .…”

“…Recall that all bounded pseudoconvex balanced domains are Bergman complete (see [11]) as well as the domain defined in (34) (see [17]).…”

“…Employing the potential theoretic methods applied by Jucha we present additionally an effective and simple algebraic criterion for a homogeneous polynomial to be square integrable on pseudoconvex balanced domains in dimension two (see Theorem 11). This theorem has many consequences as to the positive definiteness of the Bergman kernel and the positive definiteness of the Bergman metric is concerned (see Corollaries 12,13,15). In particular, with its help we may conclude the existence of domains of Siciak's type having completely different properties (see Theorem 17).…”

“…We think the methods and ideas presented in our paper may help the Reader to develop new methods to cope with the problems. Let us mention here that recently a lot of effort was invested in investigation of these problems in many classes of unbounded domains (see e. g. [2], [3], [4], [1], [15] or [16]). As we saw a problem of Wiegerinck drew a lot of effort recently and except for partial results already mentioned above the problem was repeated in a recent survey on problems in the theory of several complex variables ( [5]).…”

A Kobayashi and Bergman complete domain without bounded representations

“…3 for the definition) possesses a Bergman metric. Some other conditions (for certain unbounded X) which are sufficient for possessing a (complete) Bergman metric are also scattered in the literature, see for examples [2,8,26,28] et al…”