Deformations of annuli on Riemann surfaces and the generalization of Nitsche conjecture

Let A and A' be two circular annuli and let rho be a radial metric defined in the annulus A'. Consider the class H-rho of rho-harmonic mappings between A and A'. It is proved recently by Iwaniec, Kovalev and Onninen that, if rho = 1 (that is, if rho is Euclidean metric), then H-rho is not empty if and only if there holds the Nitsche condition (and thus is proved the J. C. C. Nitsche conjecture). In this paper, we formulate a condition (which we call rho-Nitsche conjecture) with corresponds to H rho and define rho-Nitsche harmonic maps. We determine the extremal mappings with smallest mean distortion for mappings of annuli with respect to the metric rho. As a corollary, we find that rho-Nitsche harmonic maps are Dirichlet minimizers among all homeomorphisms h : A -> A'. However, outside the rho-Nitsche condition of the modulus of the annuli, within the class of homeomorphisms, no such energy minimizers exist. This extends some recent results of Astala, Iwaniec and Martin (ARMA, 2010) where it is considered the case rho = 1 and rho = 1/vertical bar z vertical bar.