关于抛物线区域的极值拟共形映照

On extremal quasi-conformal mappings for parabolic regions

用Ωs表示抛物线区域:Ωs={z=x+iy|y>|x|s,s>0},Ws={z∈Ωs|z≠ib,b>0},在Ws上定义一个由二次微分 (这里0<α<2,k等于0或1,0≤α+k<2)所导出的Teichmuller映照,||(?)(z)||Ωs=+∞.证明了对于Ws,当s>3/1+α+k时,所给的Teichmuller映照关于其边界值是唯一极值的.而当s>1时,所给的Teichmuller映照关于其边界值是极值的,若在(?)(z)中今α=k=0或α=0,k=1则分别得到文[1]、[2]中的两个相关定理,从而本文可以看成是它们的推广.

Let Ωs indicate parabolic regions: Ωs = {z=x+iy|y>|x|s,s>0}, Ws indicate regions: Ws ={z∈Ωsz≠ib,b >0}.We define a Teichmuller mapping with quadratic differential (where 0<α<2,k = 0 or 1, 0<α+k<2)in Ws, ||(?)(z)||Ωs=+∞. In this article we have proved that above Teichmuller mapping with given boundary values is unique external for regions W , when s>3/1+α+k', and is extremal when s>1, if let α= k = 0 or α= 0, k = 1 in (?)(z) and we get separately two relative theorems of [1]and[2]. So this article may be regarded as their development.