Abstract
This paper mainly deals with the metrics and rigidity on some Hartogs-type domains in several complex variables. Firstly, we describe the Bergman kernel expansions of canonical metrics for the Hartogs domains over bounded symmetric domains, and we also obtain necessary and sufficient conditions for the canonical metrics to be balanced metrics. Secondly, we give the rigidity of proper holomorphic mappings on several Hartogs domains (i.e., Hua domains, Foc k -Bargmann-Hartogs domains and pentablock).
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