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Conformal metrics and boundary accessibility. (English) Zbl 1191.30010

A boundary point \(y\) of a domain \(\Omega\) is broadly accessible if there is a sequence of balls in \(\Omega\) such that their centers converge to \(y\) and each center can be joined to \(y\) with a path in \(\Omega\) whose length is only slightly greater than the radius of the ball.
T. E. Gerasch [Mich. Math. J. 33, 201–207 (1986; Zbl 0605.30020)] showed that, for a conformal mapping \(f : B \to \mathbb{C}\), for a.e. point \(z \in \partial B\), the radial limit \(f(z)\) is broadly accessible in \(f(B)\). The result was extended to \(\mathbb{R}^n\), \(n \geq 2\), and to quasiconformal mappings by O. Martio and R. Näkki [Bull. Lond. Math. Soc. 36, No. 1, 115–118 (2004; Zbl 1040.30010)]. This result was further extended to sets of smaller size in terms of the Hausdorff measure on \(\partial B\) by P. Koskela and S. Rohde [Math. Ann. 309, No. 4, 593–609 (1997; Zbl 0890.30013)].
The author proves a refined version of this result stated in terms of a conformal metric in the unit ball \(B^n\). For a \(K\)-quasiconformal mapping \(f : B^n \to \Omega\), the result says that, for \(s > 1\), there is a set \(E \subset \partial B^n\) with \(H^{\varphi}(E) = 0\) such that, for all \(z \in \partial B^n \setminus E \), there is a sequence of points \(y_k \to f(z) \) in \(\Omega\) so that \[ f(z) \in B^n\Big(y_k, c \delta (y_k)\big(-\log \delta (y_k)\big)^{s-1}\Big). \]
Here \(\varphi(t) = \exp\big(-(-\log t)^{1/s}\big)\), \(\delta (y) = \text{dist\,} (y, \partial \Omega)\), and \(c = c(n, s, K) > 0\). The result is essentially sharp.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30D40 Cluster sets, prime ends, boundary behavior
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References:

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