On the boundary value problem for harmonic maps of the Poincaré disc (Q1387479)

Let \(D\) be the unit disc, \(T(D)\) the space of quasisymmetric homeomorphisms \(h\) of \(\partial D\) onto itself which admit a quasiconformal extension to \(D\) and fix the points \(1,i,-1\). Let \(T_*\) be the subspace of \(T(D)\) which have a quasiconformal extension to \(D\) with complex dilation \(\mu\) satisfying \[ \iint_D \bigl| \mu(z) \bigr |^2 \rho(z) dA< \infty \] where \(\rho(z) | dz|^2\) is the Poincaré metric. Let \(B_+ \) be the Banach space of regular quadratic differentials \(\varphi dz^2\) on \(D\) with norm \[ \|\varphi \|_{B_*} =\Bigl(\iint_D \bigl| \varphi(z) \bigr|^2 \rho^{-1 }(z)dA \Bigr)^{1 /2} <\infty. \] It is shown that for any given quasisymmetric homeomorphism \(h\) of \(D\) onto itself in \(T_*\) there is a unique quasiconformal harmonic map \(f\) of \(D\) with respect to the Poincaré metric extending \(h\) and the Hopf differential \(\varphi_b= \rho(f) bz \overline b \overline z\) belongs to \(B_*\).