AI 中文总结
本文综述多复变量中Schwarzian导数在局部双全纯映射上的性质，构造不变双线性算子度量曲率，证明小界蕴含单叶性，并推广Ahlfors--Weill定理。
AI 中文摘要
在单复变量中，Schwarzian导数刻画了Möbius变换，源于二阶线性方程，并通过Nehari定理和Ahlfors--Weill定理控制单叶性。本章介绍了在$\mathbb{C}^n$中局部双全纯映射关于这三个性质已知的结果。从Oda的定义出发，我们构造了一个双线性算子，其关于Bergman度量的范数在球的自同构下不变，并证明该算子度量超平面原像的外在曲率。我们研究了由该范数有界定义的线性不变族，证明了足够小的界蕴含单叶性，并描述了取值为超平面空间的Ahlfors--Weill型延拓。还讨论了多圆柱与凸映射，并包含开放问题与练习。
英文摘要
In one complex variable the Schwarzian derivative characterizes the Möbius transformations, arises from a second-order linear equation, and controls univalence through the theorems of Nehari and of Ahlfors--Weill. This chapter presents what is known about these three properties for locally biholomorphic mappings in $\mathbb{C}^n$. Starting from Oda's definition we construct a bilinear operator whose norm, taken with respect to the Bergman metric, is invariant under the automorphisms of the ball, and we show that this operator measures the extrinsic curvature of the preimages of hyperplanes. We study the linearly invariant family defined by a bound on this norm, we prove that a sufficiently small bound implies univalence, and we describe the extension of Ahlfors--Weill type, which takes values in the space of hyperplanes. The polydisk and convex mappings are also treated, and open problems and exercises are included.
