Invariant weighted Bergman metrics on domains

TL;DR

This paper investigates conditions under which weighted Bergman metrics remain invariant under biholomorphic mappings, introducing invariant weight assignments and providing new proofs of convergence for specific kernel sequences on uniform squeezing domains.

Contribution

It introduces the concept of invariant weight assignments for weighted Bergman metrics and offers alternative proofs of convergence for Tian's and Tsuji's kernel sequences.

Findings

Bergman's minimum integral method is effective for convergence proofs.

Uniform convergence of Tian's Bergman kernel sequence is established.

Tsuji's dynamical kernel sequence also converges uniformly on uniform squeezing domains.

Abstract

In this paper, we study the cases where the weighted Bergman metrics of a domain are invariant under biholomorphisms by introducing the concept of {\it 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 uniform convergence of Tian's sequence of Bergman kernels and metrics on uniform squeezing domains. We also present a proof of the uniform convergence of Tsuji's dynamical kernel sequence on uniform squeezing domains.