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The  d -boundedness of the Kähler metric is a sufficient condition for thevanishing theorem to hold on a noncompact complete Kähler manifold, but not a necessary condition for it. Therefore, it is unknown whether the Kähler metric is  d -bounded even if thevanishing theorem holds. In this paper, we consider the  d -boundedness of the Bergman metric on a kind of Hartogs domain and give avanishing theorem on it.

The d -boundedness of the Bergman metric on a kind of Hartogs domain