Moebius映射到CAT（-1）空间的外心扩张
Circumcenter extension of Moebius maps to CAT(-1) spaces
摘要：
给定在测地完备的CAT（-1）空间$X,Y$的边界之间的Moebius同胚$f : \partial X \to \partial Y$，我们描述了$f$的一个扩张$\hat{f} : X \to Y$，称为$f$的外心映射，它是利用扩张集的外心构造的.证明了扩张$\hat{f}$与[biswas 3]中构造的$(1, \log 2)$ -拟等距扩张一致，并且是局部$1/2$ -Holder连续的.当$X,Y$是完备的，单连通流形的截面曲率$K$满足$-b^2 \leq K \leq -1$，对于某些$b \geq 1$，则扩张$\hat{f} : X \to Y$是$(1, (1 - \frac{1}{b})\log 2)$ -拟等距. Moebius映射的外心扩张对于等距合成是自然的。
英文摘要：
Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, we describe an extension $\hat{f} : X \to Y$ of $f$, called the circumcenter map of $f$, which is constructed using circumcenters of expanding sets. The extension $\hat{f}$ is shown to coincide with the $(1, \log 2)$-quasi-isometric extension constructed in [biswas3], and is locally $1/2$-Holder continuous. When $X,Y$ are complete, simply connected manifolds with sectional curvatures $K$ satisfying $-b^2 \leq K \leq -1$ for some $b \geq 1$ then the extension $\hat{f} : X \to Y$ is a $(1, (1 - \frac{1}{b})\log 2)$-quasi-isometry. Circumcenter extension of Moebius maps is natural with respect to composition with isometries.
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