HARMONIC MAPPINGS OF AN ANNULUS, NITSCHE CONJECTURE AND ITS GENERALIZATIONS By TADEUSZ IWANIEC, LEONID V. KOVALEV, and JANI ONNINEN

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 < ∞, 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 speed there corresponds an underlying differential operator which yields sharp lower bounds of R∗ r∗ in our generalization of the Nitsche Conjecture.