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

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 $D⋐{\mathbb{C}}^n$, $n\ge 2$, that admit large $\partial D$-open subsets $\mathcal{O}\subset \partial D$ such that $s_D\to 0$ approaching any point in $\mathcal{O}$. 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.

25 pages; added some expository remarks; corrected a typo in the statement of Lemma 5.1; to appear in Internat. J. Math

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