A new family of holomorphic homogeneous regular domains and some questions on the squeezing function

Abstract

We revisit the phenomenon where, for certain domains [Formula: see text], if the squeezing function [Formula: see text] extends continuously to a point [Formula: see text] with value [Formula: see text], then [Formula: see text] is strongly pseudoconvex around [Formula: see text]. In [Formula: see text], we present weaker conditions under which the latter conclusion is obtained. In another direction, we show that there are bounded domains [Formula: see text], [Formula: see text], that admit large [Formula: see text]-open subsets [Formula: see text] such that [Formula: see text] approaching any point in [Formula: see text]. This is impossible for planar domains. We pose a few questions related to these phenomena. But the core result of this paper identifies a new family of holomorphic homogeneous regular domains. We show via a family of examples how abundant domains satisfying the conditions of this result are.