Properties of squeezing functions and geometry of bounded domains

TL;DR

This paper investigates the properties and boundary behavior of squeezing functions in bounded domains, providing new estimates and applying these to various complex domains to understand their geometric and analytic characteristics.

Contribution

It introduces boundary estimates for squeezing functions using ball pinching radius and analyzes their limits and comparisons on different bounded domains.

Findings

Squeezing functions have positive lower bounds on Cartan-Hartogs domains.

Boundary estimates of squeezing functions are derived using intrinsic ball pinching radius.

Limit behavior of squeezing functions is characterized for sequences of bounded domains.

Abstract

In this article we continue the study of properties of squeezing functions and geometry of bounded domains. The limit of squeezing functions of a sequence of bounded domains is studied. We give comparisons of intrinsic positive forms and metrics on bounded domains in terms of squeezing functions. To study the boundary behavior of squeezing functions, we introduce the notions of (intrinsic) ball pinching radius, and give boundary estimate of squeezing functions in terms of these datum. Finally, we use these results to study geometric and analytic properties of some interesting domains, including planar domains, Cartan-Hartogs domains, and a strongly pseudoconvex Reinhardt domain which is not convex. As a corollary, all Cartan-Hartogs domains are homogenous regular, i.e., their squeezing functions admit positive lower bounds.