## 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