Invariant Weighted Bergman Metrics on Domains

In this paper, we study the cases where the weighted Bergman metrics of a domain are invariant under biholomorphisms by introducing the concept of invariant weight assignments, focusing on two examples by Tian and Tsuji, respectively. Using Bergman’s minimum integral method and a domain version of the Tian–Yau–Zelditch expansion for the weighted Bergman kernels and metrics, we give an alternative proof of the uniform convergence of Tian’s sequence of Bergman kernels and metrics on a uniform squeezing domian. We also present a proof of the uniform convergence of Tsuji’s dynamical kernel sequence on a uniform squeezing domian.