唯一极值映照为正则Teichmǖller映照的充要条件

Necessary and Sufficient Condition for the Uniquely Extremal Quasiconformal Mapping to be Regular Teichmuller Mapping

设 f为单位圆 D={ |z|<1}到自身 ,且与 f有相同边界值的拟共形映照类 Qf 中的唯一极值拟共形映照 ,f 的最大特征 K>1.那么 ,f 为正则 Teichmüller映照的一个充分必要条件是存在一列 Jordan曲线 γn.γn 的内部为 Dn,∪∞n=1Gn=D,且 f|γn无本质边界点 ,n=1,2 ,… .即 γn 上的每一点关于 f|γn的点特征 ,都小于从 Gn 到 f( Gn)以 f|γn为边界值的极值拟共形映照的最大特征 .

Let f with maximal dilalation K >1, be uniquely extremal in the class Q f , where Q f denotes a class of quasiconformal mappings from unit circle D={｜z ｜<1} to itself and with the same boundary values as f . Then, f will be regular Teichmüller mapping if and only if, there exists a sequence of Jordan curves γ n with ∪∞ n=1 G n=D ,and with the property that f｜ γ n has no substantial boundary point for every n .