抛物区域上拟共形映射的极值性

On the Extremality of Quasiconformal Mappings of Parabolic Domain

分别记Ω={(x,y)|y^2<4(x+1)}为平面上的抛物区域,F_K=Kx+iy+K-1/K是Ω上的水平拉伸映射,Ω=F_K(Ω),EΩΩ,Q(F_(K|E))={f:f是Ω到Ω上的拟共形映射,f|_E=F_(K|E)}.得到了F_K在Q(F_(K|E))中极值的充要条件是∞为E的聚点.

LetΩ={（x,y）｜y^2〈4（x＋1）} be parabolic domain in plane,FK=Kx＋iy＋ K-1/K be horizontal streching of Ω,Ω=FK（Ω）,EΩ Ω,Q（F（K｜E））={f：f is quasiconformal mapping of Ω onto Ω,f｜E=F（K｜E）}.It is proved that FK is extremal in Q（F（K｜E））if and only if ∞ is an accumulation piont of E.