半平面上的Chordal Loewner链与Teichmüller空间
Chordal Loewner chains and Teichmüller spaces on the half-plane
摘要：
考虑半平面上的单叶解析函数$f$满足其（预）Schwarzian导数的上确界范数在边界上为零的条件。在对$f$的某些额外假设下，证明了存在一个从$f$开始直到某个有限时间的弦Loewner链，并且这个Loewner链定义了$f$在边界上的拟共形扩张，使得它的复伸缩在边界的某个邻域中以Schwarzian导数的形式给出.这可以看作是Becker在圆盘上的相应结果的半平面形式，也是Ahlfors-Weill公式的推广。作为这种拟共形推广的应用，我们利用（预）Schwarzian导数诱导的Carleson测度消失条件，完成了半平面上VMO-Teichmüller空间元素的刻画.
英文摘要：
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.
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