Stability of conformal distortion
Let \(e^{i \theta} \mapsto e^{i \varphi (\theta)}\) be an homeomorphisms of the unit circle \(C = \{| z | = 1)\}\) such that \(\varphi (\theta)\) is a differentiable function with \(\varphi' (\theta) > 0\). Then it is well known that there exists a Jordan curve \(\Gamma\) together with conformal mappings \(F : \text{int} C \to \text{int} \Gamma, G : \text{ext} C \to \text{ext} \Gamma\) satisfying \(F(e^{i \theta)}) = G(e^{i \varphi (\theta)})\). The conformal welding (consisting of \(\Gamma\) and the pair \((F,G))\) is uniquely determined if \(G\) is normalized by \(G(z) = z + O(1/z)\) \((z \to \infty)\). The first result of the authors is that if \(\varphi\) is close to the identity in the sense that \(\varphi' (\theta)\) is close to 1 then the curve \(\Gamma\) is close to a circle in the sense that \(\Gamma\) admits some quasiconformal reflection with small maximal dilatation. This implies also that \(\Gamma\) is contained in some circular ring domain of small modulus. The main result of the paper is that under the assumptions above the problem of conformal welding is ``well posed'' which means that the solution of the problem is stable in the following sense: If \(\varphi (\theta)\) differs only a little from \(\varphi_ 0 (\theta)\) such that the ratio \(\varphi_ 0' (\theta)/ \varphi' (\theta)\) is again close to 1 then \(\Gamma\) is also close to the curve \(\Gamma_ 0\) corresponding to \(\varphi_ 0\). More precisely the distance in the Hausdorff metric is small. For the proof quasiconformal distortion theorems are used. Furthermore examples are studied showing that the quantitative estimates given in the results above are best possible in a sense. The paper closes with a result on convergence with regard to the Dirichlet norm in the special case that \(\varphi_ 0 (\theta) = \theta\), \(\Gamma_ 0 = C\).
scientific article; zbMATH DE number 720099