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

Abstract

In recent work by Zimmer it was proved that if Ω ⊂ C n is a bounded convex domain with C ∞ -smooth boundary, then Ω is strictly pseudoconvex provided that the squeezing function approaches one as one approaches the boundary. We show that this result fails if Ω is only assumed to be C 2 -smooth.

A comparison of two biholomorphic invariants

“…That this is a biholomorphic invariant follows from its definition and when D = B n , it can be checked that s D ≡ 1. Various aspects of s D have been studied of late but among those that are directly relevant to this note are its boundary behaviour on some classes of domains (for example [3], [8], [12] and [13]) and conversely, its efficacy in determining some geometric properties of the boundary of the domain if its boundary behaviour is a priori known -for example, [14] and [19].…”

Estimates for the Squeezing Function Near Strictly Pseudoconvex Boundary Points with Applications

“…E.F. Wold posed the question what is the optimal estimate for σ Ω if bΩ is C 2,ε -smooth, see [16] (see also [7]). Theorem 1 below shows that similar estimates as above hold in the C 2,ε -and C 3,ε -smooth cases.…”

Squeezing functions and Cantor sets

“…More recently, Deng-Guan-Zhang, see [2] (2012) initiated a basic study of the squeezing function. After that the squeezing function has been investigated by several authors, among them, Fornaess-Wold [7] (2015), Nikolov-Trybula-Andreev [14] (2016), Deng-Guan-Zhang [3](2016), Joo-Kim [10] (2016), Kim-Zhang [11] (2016), Zimmer [18] (2017), Fornaess-Rong [5] (2017), Fornaess-Shcherbina [6] (2017), Diederich-Fornaess [4] and Fornaess-Wold [8]. We will introduce an auxiliary function R that will enable us to bound the squeezing function from below on the limit of a certain increasing sequence of domains.…”