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As long ago as 1962 Nitsche conjectured that a harmonic homeomorphism $h\colon A(r,R) \to A(r_*, R_*)$ between planar annuli exists if and only if$\frac{R_*}{r_*} \ge {1/2} (\frac{R}{r} + \frac{r}{R})$. We prove thisconjecture when the domain annulus is not too wide; explicitly, when $\log\frac{R}{r} \le {3/2}$. For general $A(r,R)$ the conjecture is proved underadditional assumption that either $h$ or its normal derivative have vanishingaverage on the inner boundary circle. This is the case for the critical Nitschemapping which yields equality in the above inequality. The Nitsche mappingrepresents so-called free evolution of circles of the annulus $A(r,R)$. It willbe shown on the other hand that forced harmonic evolution results in greaterratio $\frac{R_*}{r_*}$. To this end, we introduce the underlying differentialoperators for the circular means of the forced evolution and use them to obtainsharp lower bounds of $\frac{R_*}{r_*}$.

Harmonic mappings of an annulus, Nitsche conjecture and its generalizations