Holomorphic Bergman isometries between bounded domains and the extension of Lu's theorem

Abstract

We study holomorphic maps between complex manifolds that preserve the Bergman metric up to a positive constant. In the equal dimensional case, either the source is a Stein manifold and the target is a bounded domain in the complex Euclidean space, or the source is a bounded pseudoconvex domain and the target is a complex manifold, holomorphic Bergman isometry is a biholomorphism onto the complement of a relatively closed locally pluripolar set, and the constant is necessarily one. These include the bounded domains in the complex Euclidean space as special cases. Some applications to curvature-preserving maps are obtained. We further prove the rigidity result for local holomorphic Bergman isometries into Cartesian product of strongly pseudoconvex domains, with applications to finite analytic and modular correspondences.