Harmonic mapping problem in the plane (Q648525)

This is a survey article of work done by the authors and their collaborators. The subject matter concerns the notions of the conformal modulus and the affine modulus of a doubly connected plane domain, Nitsche's conjecture on the Harmonic Mapping Problem, and Hopf differentials and diffeomorphic approximation of Sobolev homeomorphisms. The main sources for these results are: [\textit{T. Iwaniec, L. V. Kovalev} and \textit{J. Onninen}, J. Am. Math. Soc. 24, No. 2, 345--373 (2011; Zbl 1214.31001); ``Doubly connected minimal surfaces and extremal harmonic mappings'', \url{arXiv:0912.3542}; ``Harmonic mapping problem and affine capacity'', \url{arXiv:1001.2124}; ``Hopf differentials and smoothing Sobolev homeomorphisms'', \url{arXiv:1006:5174}] and [\textit{N.-T. Koh} and \textit{L. V. Kovalev}, Bull. Lond. Math. Soc. 43, No. 1, 91--96 (2011; Zbl 1210.31001)].