Moduli of doubly connected domains under univalent harmonic maps (Q2117502)

Let \(\mathcal{T}(t)=\mathbb{C}\setminus([-1,1]\cup[t,\infty))\), \(t>1\), be a Teichmüller doubly connected domain. The Teichmüller-Nitsche problem is formulated as follows: For which values \(s,t\), \(1<s,t<\infty\), does a harmonic homeomorphism \(f:\mathcal{T}(s)\rightarrow\mathcal{T}(t)\) exist? In the paper, the Teichmüller-Nitsche problem is solved for symmetric harmonic homeomorphisms between \(\mathcal{T}(s)\) and \(\mathcal{T}(t)\). This problem is solved by using the method of extremal length. The following question suggested by \textit{T. Iwaniec} et al. [Proc. R. Soc. Edinb., Sect. A, Math. 141, No. 5, 1017--1030 (2011; Zbl 1267.30059)] is also considered: Characterize pairs \((\Omega,\Omega^*)\) of doubly connected domains that admit a univalent harmonic mapping from \(\Omega\) onto \(\Omega^*\). This question is tested regarding the moduli of the doubly connected domains related by harmonic homeomorphisms. The paper concludes with relevant questions.