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

The d-boundedness of the Kähler metric is a sufficient condition for the L2 vanishing 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 the L2 vanishing theorem holds. In this paper, we consider the d-boundedness of the Bergman metric on a kind of Hartogs domain and give a L2 vanishing theorem on it.