Harmonic mappings of an annulus, Nitsche conjecture and its generalizations

\def\xxlong{\mathop{{\rm onto}\atop \longrightarrow}\nolimits} As long ago as 1962 Nitsche conjectured that a harmonic homeomorphism $h \colon \ A(r,R) \xxlong A(r_\ast, R_\ast)$ between planar annuli exists if and only if $ \frac{R_\ast}{r_\ast} \ge \frac{1}{2} \left(\frac{R} {r} + \frac{r}{R}\right)$. We prove this conjecture when the domain annulus is not too wide; explicitly, when $R \le e^{3/2} r$. We also treat the general annuli $A(r,R)$, $ 0 0$r$R\infty$, 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 $ \frac{R_\ast}{r_\ast} = \frac{1}{2} \left(\frac{R}{r} + \frac{r}{R}\right)$ 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 $\frac{R_\ast}{r_\ast}$ in the deformed (target) annulus. To every initial speed there corresponds an underlying differential operator which yields sharp lower bounds of $\frac{R_\ast}{r_\ast}$ in our generalization of the Nitsche Conjecture.