论文标题
半平面上的和弦洛夫纳连锁店和Teichmüller空间
Chordal Loewner chains and Teichmüller spaces on the half-plane
论文作者
论文摘要
我们考虑在半平面上的单个分析函数$ f $，这满足了其（前）Schwarzian衍生产品在边界上消失的最高规范的条件。在$ f $上的某些额外假设下，我们表明存在一个从$ f $到某个有限时间发起的和弦loewner链，而该洛伊纳链定义了$ f $的准符号扩展，以在边界上进行$ f $，以使其复杂的扩张在某个社区中明确地以（pre-）schwarzian衍生作用在某个社区中给出。这可以被视为Becker在磁盘上开发的相应结果的半平面版，也可以将AHLFORS-WEILL公式的概括。作为此准文献扩展的应用，我们使用（前）Schwarzian衍生物诱导的消失的Carleson测量条件来完成半平面上VMO-TeichMüller空间的元素的表征。
We consider a univalent analytic function $f$ on the half-plane satisfying the condition that the supremum norm of its (pre-)Schwarzian derivative vanishes on the boundary. Under certain extra assumptions on $f$, we show that there exists a chordal Loewner chain initiated from $f$ until some finite time, and this Loewner chain defines a quasiconformal extension of $f$ over the boundary such that its complex dilatation is given explicitly in terms of the (pre-)Schwarzian derivative in some neighborhood of the boundary. This can be regarded as the half-plane version of the corresponding result developed on the disk by Becker and also the generalization of the Ahlfors-Weill formula. As an application of this quasiconformal extension, we complete the characterization of an element of the VMO-Teichmüller space on the half-plane using the vanishing Carleson measure condition induced by the (pre-)Schwarzian derivative.
