Hyperbolically convex constricted domains (Q961063)

Let \(\mathbb D\) be the unit disk in the complex plane \(\mathbb C\), and let \(\mathbb T=\partial\mathbb D\). A subdomain \(\mathcal C\) of \(\mathbb D\) is called hyperbolically convex (in short h-convex) if the h-segment between any two points in \(\mathcal C\) also lies in \(\mathcal C\). The authors introduce a concept of constricted domains relative to the hyperbolic geometry of \(\mathbb D\). A domain \(\mathcal C\subset\mathbb D\) is said to be constricted if there are sequences \(a_n\) and \(b_n\) of points on \(\tilde\partial\mathcal C\), h-discs \(D_n\) with h-center \(z_n\) and h-radius \(r_n\leq1\), and h-lines \(L_n\) such that \(a_n\) and \(b_n\) are on different sides of \(L_n\), \(z_n\in L_n\cap D_n\subset \mathcal C\) and \[ \lim\limits_{n\to\infty}\frac{d(a_n,z_n)}{r_n}= \lim\limits_{n\to\infty}\frac{d(b_n,z_n)}{r_n}=0\,, \] where \(\tilde\partial\mathcal C\) is the hyperbolic boundary of \(\mathcal C\) in \(\mathbb D\), and \(d\) is hyperbolic distance in any model of the hyperbolic plane. The authors prove that constricted domains cannot be quasidisks. It turns out that the converse is also true for h-convex domains: Let \(\mathcal C\) be an h-convex domain in \(\mathbb D\) which is not a quasidisk. Then \(\mathcal C\) is a constricted domain. This is the main result of the paper. If \(z_0\) is a limit point of the sequence \(z_n\) in the definition of a constricted domain, then it is also a limit point of the sequences \(a_n\) and \(b_n\), and therefore \(z_0\in\partial\mathcal C\). In this case, \(z_0\) is called a pinch of \(\mathcal C\). The authors show that pinches can only occur in \(\mathbb T\), and therefore every h-convex domain \(\mathcal C\) with \(\overline{\mathcal C}\subset\mathbb D\) is a quasidisk. The authors present a classification of pinches. The idea behind this classification is to distinguish those pinches \(z_0\) that are limit points of sequences of points \(z_n\) which can be chosen to lie on a single h-line. A pinch of this type is called regular, otherwise it is called irregular. An interesting fact is that if an a h-convex domain is constricted at a point \(z_0\), and if it is locally contained in a Stolz angle at \(z_0\), then \(z_0\) must be a regular pinch. However, this property does not characterize regular pinches as shown by an example.