A characterization of domains of holomorphy by means of their weighted Skwarczyński distance
Zbigniew Pasternak-Winiarski , Paweł Wójcicki
AbstractM. Skwarczyński (†) introduced pseudodistance on domains in Cn which under some conditions (if the domain is bounded for instance) gives rise to biholomorphically invariant distance, i.e., invariant under biholomorphic transformations. One can find a proof that completeness with respect to Skwarczyński distance implies completeness with respect to Bergman distance, which implies that the considered domain is a domain of holomorphy. In this paper we give a characterization of domains of holomorphy with the help of a weighted version of Skwarczyński pseudodistance. We will work with a special kind of weights, called “admissible weights”. Midway, we obtain a new proof (even in the unweighted case) of the theorem that the so-called Kobayashi condition implies Bergman completeness, which may be helpful in answering the (open) question if Bergman completeness and Skwarczyśnki completeness are equivalent or not.
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