环上的调和映射，Nitsche猜想及其推广
Harmonic mappings of an annulus, Nitsche conjecture and its generalizations
摘要：
早在1962年，Nitsche就证明了平面环之间存在调和同胚$h \colon A(r,R) \to A(r_*, R_*)$当且仅当$\frac{R_*}{r_*} \ge {1/2} (\frac{R}{r} + \frac{r}{R})$。我们证明这个猜想时，域环不是太宽，明确地说，当$\log \frac{R}{r} \le {3/2}$。对于一般的$A(r,R)$，在附加假设$h$或其正规导数在内边界圆上的平均为零的条件下，证明了该猜想.这是临界Nitsche映射的情况，它在上面的不等式中产生相等。Nitsche标测代表了所谓的瓣环$A(r,R)$圆的自由演化。另一方面，它将表明，强制谐波演变的结果在更大的比率$\frac{R_*}{r_*}$。为此，我们引入了基本的微分算子的强制进化的循环手段，并使用它们来获得尖锐的下界$\frac{R_*}{r_*}$。
英文摘要：
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 this conjecture 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 under additional assumption that either $h$ or its normal derivative have vanishing average on the inner boundary circle. This is the case for the critical Nitsche mapping which yields equality in the above inequality. The Nitsche mapping represents so-called free evolution of circles of the annulus $A(r,R)$. It will be shown on the other hand that forced harmonic evolution results in greater ratio $\frac{R_*}{r_*}$. To this end, we introduce the underlying differential operators for the circular means of the forced evolution and use them to obtain sharp lower bounds of $\frac{R_*}{r_*}$.
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