Moduli of Doubly Connected Domains Under Univalent Harmonic Maps

Iwaniec et al. (Proc R Soc Edinb 141A:1017–1030, 2011) raised the following problem: For which values s , t , 1 < s , t < ∞ , does there exist a harmonic homeomorphism f : T ( s ) → T ( t ) , where T ( . ) is a Teichmüller domain? By restricting ourselves to harmonic homeomorphisms symmetric about the real axis, we establish a two-fold purpose: (a) solve this problem by using the theory of extremal length and (b) test Conjecture 1.4 of loc. cit. regarding the moduli of the doubly connected domains related by harmonic homeomorphisms in light of our results. The paper concludes with relevant and interesting questions.