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

TL;DR

This paper investigates the boundary behavior of the squeezing function on complex domains, introduces a new family of holomorphic homogeneous regular domains, and explores their geometric properties and implications.

Contribution

It identifies a new family of holomorphic homogeneous regular domains and demonstrates their abundance through examples, advancing understanding of boundary regularity and the squeezing function.

Findings

Boundary points with squeezing function value 1 imply strong pseudoconvexity under certain conditions.

Existence of bounded domains with large boundary subsets where the squeezing function tends to zero.

Introduction of a new family of holomorphic homogeneous regular domains.

Abstract

We revisit the phenomenon where, for certain domains $D$, if the squeezing function $s_D$ extends continuously to a point $p\in \partial{D}$ with value $1$, then $\partial{D}$ is strongly pseudoconvex around $p$. In $\mathbb{C}^2$, we present weaker conditions under which the latter conclusion is obtained. In another direction, we show that there are bounded domains $D\Subset \mathbb{C}^n$, $n\geq 2$, that admit large $\partial{D}$-open subsets $\mathscr{O}\subset \partial{D}$ such that $s_D\to 0$ approaching any point in $\mathscr{O}$. 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.