Estimate of the squeezing function for a class of bounded domains

Abstract

Abstract. We construct a class of bounded domains, on which the squeezing function is not uniformly bounded from below near a smooth and pseudoconvex boundary point.

On Boundary Points at Which the Squeezing Function Tends to One

“…On the other hand, as shown recently in [7], the answer to Question 1 is negative if no other conditions on the bounded domain than the boundary being C 2 are assumed. Also the high dimensions can imply some unexpected phenomenon [6]. Thus the hypothesis of the our theorem is in some sense reasonable.…”

Infinite Dimensional Holomorphic Homogeneous Regular Domains

“…[21, 11.4, 37.7]). We note that a domain of holomorphy need not be HHR even in finite dimensions, as shown in [8,Theorem 1].…”

“…These manifolds possess many important geometric properties (e.g. all classical metrics on them are equivalent) [19,20] and have also been studied by several authors (see, for example, [6,7,8,14,24]) in the case of complex domains. In particular, it has been shown in [24] that a holomorphic homogeneous regular bounded domain D in C n must be pseudoconvex and all strongly convex domains in C n are holomorphic homogeneous regular.…”

A non-strictly pseudoconvex domain for which the squeezing function tends to 1 towards the boundary

“…Yeung [8] showed that the answer is yes for strongly convex domains in C n , and Kim-Zhang [6] and Deng-Guan-Zhang [3] showed that the answer is yes for strictly pseudoconvex domains. On the other hand, Fornaess-Rong [4] showed that the answer is no for n ≥ 3.…”

“…For further results about the squeezing function the reader may also consult the references [1], [2], [3], [4], [5], [6], [7], [8], [9]. In the last section we will post some open problems.…”