The squeezing function: exact computations, optimal estimates, and a new application

TL;DR

This paper introduces new methods for computing and estimating the squeezing function, applying it to identify non-product domains, compute exact values for symmetric domains, and explore their geometric properties.

Contribution

It provides a unified approach to compute and estimate the squeezing function, extending previous results to product and exceptional domains, and introduces a new application for domain classification.

Findings

Exact squeezing function values for bounded symmetric domains.

Optimal estimates for squeezing functions of specific domains.

Identification of holomorphic homogeneous regular domains.

Abstract

We present a new application of the squeezing function $s_D$, using which one may detect when a given bounded pseudoconvex domain $D\varsubsetneq \mathbb{C}^n$, $n\geq 2$, is not biholomorphic to any product domain. One of the ingredients used in establishing this result is also used to give an exact computation of the squeezing function (which is a constant) of any bounded symmetric domain. This extends a computation by Kubota to any Cartesian product of Cartan domains at least one of which is an exceptional domain. Our method circumvents any case-by-case analysis by rank and also provides optimal estimates for the squeezing functions of certain domains. Lastly, we identify a family of bounded domains that are holomorphic homogeneous regular.