## A characterization of domains of holomorphy by means of their weighted Skwarczyński distance

### Zbigniew Pasternak-Winiarski , Paweł Wójcicki

#### Abstract

M. 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.
 Author Pages 375-384 Publication size in sheets 0.5 Book Kielanowski Piotr, Ali S. Twareque, Bieliavsky Pierre, Odzijewicz Anatol, Schlichenmaier Martin, Voronov Theodore (eds.): GEOMETRIC METHODS IN PHYSICS, Trends in Mathematics, vol. 6, 2016, SPRINGER INT PUBLISHING AG, ISBN 978-3-319-31755-7 Keywords in English Weighted Bergman kernel, admissible weight, sequence of domains Abstract in Polish W pracy przedstawiona została charakteryzacja obszarów holomorficzności przy pomocy ważonych jąder Bergmana. Dodatkowo, zestawienie dobrze znanych wyników pozwoliło uzyskać nowy dowód twierdzenia, że tzw. warunek Kobayashi implikuje zupełność w sensie Bergmana. DOI DOI:10.1007/978-3-319-31756-4_29 URL http://link.springer.com/chapter/10.1007%2F978-3-319-31756-4_29 Language en angielski Score (nominal) 15 Score Ministerial score = 15.0, 27-06-2017, BookChapterSeriesAndMatConfMinisterial score (2013-2016) = 15.0, 27-06-2017, BookChapterSeriesAndMatConf Citation count* 0
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