Curvature and proper holomorphic mappings between bounded domains in C ⁿ

In this paper we discuss some connections between proper holomorphic mappings between domains in C" and the boundary behaviors of certain canonical invariant metrics (Cheng-Yau-Einstein Kahler metric, Bergman metric, intrinsic measures, etc.). Some compactness theorems have been proved (Theorem 4, Theorem 5). This generalizes an earlier result proved by the second author.