Carathéodory and Smirnov type theorems for harmonic mappings of the unit disk onto surfaces (Q2860884)

The authors prove representation theorems which are versions of Smirnov's theorem and Carathéodory type theorems for harmonic homeomorphisms of the unit disk onto Jordan surfaces with rectifiable boundaries. Let \(\Sigma\subseteq{\mathbb R}^n\) be a closed Jordan surface with rectifiable boundary \(\Gamma\), \(\Sigma_o=\Sigma\setminus\Gamma\) and \(f:{\mathbf U}\rightarrow\Sigma_o\) a harmonic homeomorphism of the unit disk \({\mathbf U}\subset\mathbb C\) onto \(\Sigma_o\). Then there exists a function \(\phi:[0,2\pi]\rightarrow\Gamma\), \(\phi(0)=\phi(2\pi)\) with bounded variation and with at most a countable set of points of discontinuity, where it has left and right limits, such that \(f=\text{P}[\phi]\). Here \(\text{P}[\phi]\) is the Poisson integral of \(\phi\). If \(\Gamma\) does not contain any segment, then \(f\) has a continuous extension up to the boundary. If \(\theta_0\in[0,2\pi]\) is a point of discontinuity of \(\phi\), then there exist \(A_0=\lim_{t\uparrow\theta_0}\phi(t)\), \(B_0=\lim_{t\downarrow\theta_0}\phi(t)\) and \(C_{\mathbf U}(f,e^{i\theta_0})=[A_0,B_0]\subseteq\Gamma\). \(C_{\mathbf U}(f,e^{\zeta})\) is the cluster set. The authors establish some classical inequalities for harmonic surfaces. For example, the Riesz-Zygmund inequality for harmonic surfaces. Let \(\Sigma\subseteq{\mathbb R}^n\) be a harmonic Jordan surface which is spanned by a rectifiable curve \(\Gamma\) and parametrized by harmonic coordinates \(\tau\). Then for every \(s\in[0,2\pi]\) NEWLINE\[NEWLINE \int_{-1}^1|\partial_r\tau(re^{is})|\,dt\leqslant\frac12|\Gamma|\,. NEWLINE\]