Length of curves under conformal mappings
Let f be a conformal map of the unit disk D onto a region \(\Omega\). Hayman and Wu proved that there is an absolute constant A such that for any line L it follows that \[ (*)\quad \lambda (f^{-1}(\Omega \cap L))<A. \] (\(\lambda\) (E) denotes the Hausdorff linear measure of the set E.) Garnett, Gehring and Jones gave a simpler proof of this result and considered conditions on the rectifiable Jordan curve in order that (*) continues to hold. The authors prove that (*) remains valid for quasismooth, that is, chord-arc, curves with a constant A that depends only on the chord-arc constant. In addition, they prove that \(\lambda (f^{-1}(\Omega \cap L))<\infty\) if \(\Omega\) is a hyperbolic plane region, f:D\(\to \Omega\) is a universal covering map, L is a quasismooth arc and \(\Omega\) is of finite length type. This latter condition means that \(\sum \lambda (\partial T{\mathcal D})<\infty\), where \({\mathcal D}\) denotes the Dirichlet fundamental region for the covering projection and the sum is over all T in the group of cover transformations. In particular, this result holds for any finitely connected hyperbolic region without point boundary components. The tools used in the proofs include harmonic measure, quasiconformal mappings and Carleson measures.
scientific article; zbMATH DE number 3987622