共形自然扩张再探
Conformally Natural extensions revisited
摘要：
本文重新讨论了Douady和Earle在1986年《数学学报》第157卷中定义的$\SS^n$上测度的共形重心的概念。目的是将有理映射从黎曼球面$\Cbar\isom\SS^2$推广到（双曲）三球$\BB^3$，从而通过反射推广到$\SS^3$。由Douady和Earle在同胚的情况下开创的构造实际上给出了更一般的映射的扩展，例如$\C\subset\Cbar$上的整个超越映射。它适用于任何维度的地图。
英文摘要：
In this note we revisit the notion of conformal barycenter of a measure on $\SS^n$ as defined by Douady and Earle in Acta Math. Vol 157, 1986. The aim is to extend rational maps from the Riemann sphere $\Cbar\isom\SS^2$ to the (hyperbolic) three ball $\BB^3$ and thus to $\SS^3$ by reflection. The construction which was pioneered by Douady and Earle in the case of homeomorphisms actually gives extensions for more general maps such as entire transcendental maps on $\C\subset\Cbar$. And it works for maps in any dimension.
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