Properties of squeezing functions and global transformations of bounded domains

Abstract

The central purpose of the present paper is to study boundary behaviors of squeezing functions on some bounded domains. We prove that the squeezing function of any strongly pseudoconvex domain tends to 1 near the boundary. In fact, such an estimate is proved for the squeezing function on any bounded domain near its globally strongly convex boundary points. We also study the stability properties of squeezing functions on a sequence of bounded domains, and give some comparisons of intrinsic measures and metrics on bounded domains in terms of squeezing functions. As applications, we give new proofs of several well-known results about geometry of strongly pseudoconvex domains, and prove that all Cartan-Hartogs domains are homogenous regular. Finally, some related problems for further study are proposed.