A problem on extremal quasiconformal extensions (Q2644442)

The author gives a short survey on a problem on extremal quasiconformal extensions. Denote the unit disk and the unit circle in the complex plane by \(\Delta\) and \(\Gamma\) respectively. Let \(h\) be a sense-preserving quasisymmetric mapping of \(\Gamma\) onto itself. It is well known that there exist quasiconformal extensions of \(h\) onto \(\Delta\). Define \[ K_1(h)=\inf\{K: \text{\(h\) has a \(K\)-quasiconformal extension to a self map of \(\Delta\)\}}. \] Let \(z_1\), \(z_2\), \(z_3\) and \(z_4\) be four points on \(\Gamma\) in the positive direction. Then they determine a unique topological quadrilateral with domain \(\Delta\) and vertices \(z_1\), \(z_2\), \(z_3\) and \(z_4\) which we denote by \(\mathcal Q=\Delta( z_1, z_2, z_3, z_4)\). Denote the conformal modulus of \(\mathcal Q\) by \(M(\mathcal Q)\). Similarly, denote \(h(\mathcal Q)=\Delta(h(z_1),h(z_2),h(z_3),h(z_4))\) and its conformal modulus by \(M(h(\mathcal Q))\). Define \[ K_0(h)=\sup\{M(h(\mathcal Q))/M(\mathcal Q): \text{\(\mathcal Q\) is a topological quadrilateral with domain \(\Delta\)}\} \] and \[ H(h)=\inf\{K:\text{\(h\) has a \(K\)-quasiconformal extension \(f\) to \(\Delta_r\)}\}, \] where \(\Delta_r=\{z: r< | z| < 1 \}\). If there exists a non-degenerated quadrilateral \(\mathcal Q\) such that \(K_0(h)=M(h(\mathcal Q)/M(\mathcal Q)\), we adopt the notation \(K_0^q(h)\) instead of \(K_0(h)\). It is interesting to study the relationships between \(K_0(h)\), \(K_1(h)\) and \(H(h)\). In fact, it had been an open problem for a long time to determine whether or not the equality \(K_0(h)=K_1(h)\) always holds before J.~M.~Anderson and A.~Hinkkanen in 1995 disproved this by constructing concrete examples of a family of affine mappings of some parallelograms. The main results of the paper are as follows: (i) If \(K^q_0 (h) = K_0 (h)\), then there exists a non-degenerated quadrilateral \(\mathcal Q\) so that \(K_0(h)=M(h(\mathcal Q))/M(\mathcal Q)\). Vice versa, if \(K_0^q(h)\neq K_0(h)\), then there exists a sequence of degenerating quadrilaterals \(\{\mathcal Q_n\}\) so that \[ K_0(h)=\lim_{n\to\infty} M(h(\mathcal Q_n))/M(\mathcal Q_n). \] (ii) If \(1\leq H(h)\leq K^\ast\) or \(K^\ast<H(h)<K_1(h)\) then \(K_0(h)<K_1(h)\), where \(K^\ast\) is the Strebel number. (iii) Suppose that \(K_0(h)=K_1(h)\). Then \(H(h)\) is not necessarily equal to \(K_1(h)\). (iv) Suppose that \(H(h)=K_1(h)\). Then \(K_0(h)\) is not necessarily be equal to \(K_1(h)\).