Harmonic mappings of an annulus,nitsche conjecture and its generalizations

As long ago as 1962 Nitsche conjectured that a harmonic homeomorphism h: A(r, R) onto→A(r*, R*) between planar annuli exists if and only if R*r*≥ 1/2(R/r+ r/R). We prove this conjecture when the domain annulus is not too wide; explicitly, when R ≤ e3/2r. We also treat the general annuli A(r, R), 0 < r < R < 8, and obtain the sharp Nitsche bound under additional assumption that either h or its normal derivative have vanishing average along the inner circle of A(r, R). We consider the family of Jordan curves in A(r*, R*) obtained as images under h of concentric circles in A(r, R). We refer to such family of Jordan curves as harmonic evolution of the inner boundary of A(r, R). In the borderline case R*r*= 1/2(R/r+ r/R)the evolution begins with zero speed. It will be shown, as a generalization of the Nitsche Conjecture, that harmonic evolution with positive initial speed results in greater ratio R*r*in the deformed (target) annulus. To every initial isnpeoeudr tgheenreercaolirzraetsipoonnodfs tahne uNnidtsecrlhyeinCgodnijfefcetruernet.ial operator which yields sharp lower bounds of R* in our generalization of the Nitsche Conjecture.