Higher Bers地图
Higher Bers maps
摘要：
Bers嵌入将富克斯群$G$的Teichmüller空间实现为$G$的有界二次微分的复巴拿赫空间的开的、有界的和可收缩的子集。它利用了Teichmüller空间的schlicht模型，其中每个点都由圆盘上的单射全纯函数表示，并且通过Schwarzian微分算子构造映射。本文证明了一类作用于圆盘上的函数的微分算子诱导Teichmüller空间的全纯映射，并得到了诱导映射在原点处的微分的一般公式.这项工作的主要重点，但是，是在两个特定系列的这种映射，被称为高Bers地图，因为它们是由所谓的高Schwarzians-推广的经典Schwarzian算子。对于这些地图，我们证明了几个进一步的结果。最后一节讨论了可能的应用，开放的问题和猜测。
英文摘要：
The Bers embebbing realizes the Teichmüller space of a Fuchsian group $G$ as a open, bounded and contractible subset of the complex Banach space of bounded quadratic differentials for $G$. It utilizes the schlicht model of Teichmüller space, where each point is represented by an injective holomorphic function on the disc, and the map is constructed via the Schwarzian differential operator. In this paper we prove that a certain class of differential operators acting on functions of the disc induce holomorphic mappings of Teichmüller spaces, and we also obtain a general formula for the differential of the induced mappings at the origin. The main focus of this work, however, is on two particular series of such mappings, dubbed higher Bers maps, because they are induced by so-called higher Schwarzians -- generalizations of the classical Schwarzian operator. For these maps, we prove several further results. The last section contains a discussion of possible applications, open questions and speculations.
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