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Quasihyperbolic geodesics in John domains. (English) Zbl 0702.30007

Suppose that D is a proper subdomain of \({\mathbb{R}}^ n\), that \(x_ 0,x_ 1\in D\) and that \(b\geq 1\). A rectifiable arc \(\gamma\) is said to be a b- cone arc from \(x_ 1\) to \(x_ 0\) if \(\gamma\) joins \(x_ 1\) to \(x_ 0\) in D and if \(\ell (\gamma (x_ 1,x))\leq bd(x,\partial D)\) for all \(x\in \gamma\). Here \(\ell (\gamma (x_ 1,x))\) is the Euclidean length of the arc \(\gamma (x_ 1,x)\) from \(x_ 1\) to x and d(x,\(\partial D)\) is the Euclidean distance from x to \(\partial D\). The domain D is said to be a b-John domain with center \(x_ 0\) if for each \(x_ 1\in D\) there is a b-cone arc from \(x_ 1\) to \(x_ 0.\)
The authors address the following question: Suppose that D is a c-John domain with center \(x_ 0\) and that \(\gamma\) is a quasihyperbolic geodesic which joins \(x_ 1\in D\) to \(x_ 0\). Is \(\gamma\) a b-cone arc for some \(b=b(c)?\) The authors show that the answer is yes when \(n=2\) and D is simply connected, and in general no when \(n>2\) or D is multiply connected. They also provide a new characterization of simply connected John domain, which is reminiscent of the Ahlfors criterion for quasicircles: A bounded simply connected plane domain D is a John domain if and only if there exists a constant a such that for each crosscut \(\alpha\) of D, \[ \min (diam(D_ 1),diam(D_ 2))\leq a \dim (\alpha), \] where \(D_ 1\) and \(D_ 2\) are the components of \(D\setminus \alpha\).
Reviewer: D.Lesley

MSC:

30C20 Conformal mappings of special domains
30C62 Quasiconformal mappings in the complex plane
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
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