Quadrilaterals and extremal quasiconformal extensions
If \(f\), a sense-preserving homeomorphism of the boundary \(\partial D\) of the unit disc \(D\), admits a (continuous) \(K\)-quasiconformal extension to \(D\), for the module \(M(Q)\) of any quadrangle \(Q\) with domain \(D\) and the corresponding module \(M(f(Q))\) we have \(M(f(Q))/M(Q)\leq K\) (only a one-sided inequality need to be stated because of the conjugate problem). Let \(K_0= K_0(f)\) be the least upper bound of such ratio for all such \(Q\). Let \(K_1= K_1(f)= \text{g.l.b.} K\) for all quasiconformal extensions. Evidently \(K_0\leq K_1\) on the other hand consideration of the Beurling-Ahlfors necessary and sufficient conditions for the existence of quasiconformal extensions shows that when \(K_0\) is finite there will be a quasiconformal extension with maximal dilation bounded by some function of \(K_0\). It has been conjectured by Yang that always \(K_0= K_1\). This paper shows that this conjecture is false. The basic idea of the proof is simple. The parallelogram \(V\) with vertices \(0, 1, \alpha+ 1+ i\beta\), \(\alpha+i\beta\), \(\alpha> 0\), \(\beta> 0\), is mapped by the affine mapping \(F_k\) taking \(x+ iy\) to \(Kx+ iy\) \((K> 1)\) on the parallelogram \(F_k(V)\) and this mapping is uniquely extremal quasiconformal for its boundary values with maximal dilation \(K\). If \(\Phi_j\), \(j= 1, 2\), map \(V\), \(F_k(V)\) conformally on the unit disc with homeomorphic extension to the boundaries, \(\widetilde F_k= \Phi_2 F_k \Phi^{- 1}_1\) maps \(D\) quasiconformally onto itself with \(\partial D\) mapped homeomorphically and \(K_1(\widetilde F_k)=K\). It is then shown that \(K_1(\widetilde F_k)< K\). If we had \(K_0(\widetilde F_k)= K\) there would be basically two possibilities. Either \(K_0(\widetilde F_k)\) would be a maximum attained for some quadrangle \(Q\) and its image or it would not be and there would exist quadrangles \(Q_n\) with \(\lim_{n\to \infty} m(\widetilde F_n(Q_n))/m(Q_n)= K_0(\widetilde F_k)\). The authors consider the equivalent problem in the context of the parallelograms and obtain a contradiction. In the first case there is a simple proof. In the second case the vertices of a subsequence of the quadrangles, properly enumerated, would converge to four not necessarily distinct points but these could not be the vertices of a quadrangle so that there are various types of degeneracy. This requires rather complicated consideration of a number of different possibilities. In fact some cases are ``left to the reader''. As the authors point out to the reviewer [Indiana Univ. Math. J. 37, 929-934 (1988; Zbl 0648.30019)] has given a \(K\)-specific necessary and sufficient condition for the existence of a quasiconformal extension using polygons and more general modules of curve families.
scientific article; zbMATH DE number 838581