Tree-like decompositions of simply connected domains
A bounded domain \(\Omega\) in the plane is called a Lipschitz crescent if there are \(\varepsilon>0\) and \(\theta\in(0,\pi/2)\) so that \(\partial\Omega\) consists of two arcs connecting \(-1\) to \(+1\); the first a circular arc in the upper half-plane that makes angle \(\theta\) with the real line at \(\pm1\) and the second is a Lipschitz graph for which the slopes are bounded above by \(\theta-\varepsilon\) and below by \(-\varepsilon\). Any bounded Möbius image of such a domain is also called a Lipschitz crescent. The main result of the paper under review is as follows. Theorem 1.1. There is an \(M<\infty\) so that every simply connected plane domain \(\Omega\) has a collection of disjoint circular arc crosscuts \(\Gamma=\bigcup \gamma_k \) with \(\sum_k l(\gamma_k)\leq M l(\partial\Omega)\) and so that each connected component of \(\Omega\backslash\Gamma\) is an \(M\)-Lipschitz crescent. Theorem 1.1 is a strengthening of a theorem of \textit{P. W. Jones} who proved in [Invent. Math. 102, No. 1, 1--15 (1990; Zbl 0731.30018)] that every simply connected plane domain \(\Omega\) has a decomposition into Lipschitz domains \(\Omega_k\), such that \[ \sum_k l(\partial\Omega_k)= O(l(\partial\Omega)). \]
scientific article; zbMATH DE number 6011512
scientific article