Properties of squeezing functions and global transformations of bounded domains

TL;DR

This paper investigates the boundary behavior of squeezing functions on bounded domains, showing they tend to 1 near strongly pseudoconvex boundaries, and explores their stability and applications to domain geometry.

Contribution

It provides new results on the boundary limits of squeezing functions, their stability, and applications to geometric properties of complex domains, including Cartan-Hartogs domains.

Findings

Squeezing functions tend to 1 near strongly pseudoconvex boundary points.

Stability of squeezing functions on sequences of bounded domains.

All Cartan-Hartogs domains are homogenous regular.

Abstract

The central purpose of the present paper is to study boundary behavior of squeezing functions on bounded domains. We prove that the squeezing function of a strongly pseudoconvex domain tends to 1 near the boundary. In fact, such an estimate is proved for the squeezing function on any domain near its globally strongly convex boundary points. We also study the stability of squeezing functions on a sequence of bounded domains, and give comparisons of intrinsic measures and metrics on bounded domains in terms of squeezing functions. As applications, we give new and simple 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 that ask for further study are proposed.