论文标题
Busemann功能和Gromov双曲空间的均匀化
Busemann functions and uniformization of Gromov hyperbolic spaces
论文作者
论文摘要
Bonk，Heinonen和Koskela研究的Gromov倾斜空间的统一理论概括了将经典的Poincaré球类型模型用作起点的情况。在本文中，在基础域是无限的情况下，我们开发了这种方法，与经典的庞加莱半空间模型相对应。更确切地说，我们通过Busemann功能在Gromov双曲线空间上研究保形密度，并证明变形空间是无限的均匀空间。此外，我们表明，在格罗莫夫边界上的一个点和无绑定的局部紧凑型统一空间的近距离的bilipschitz类别之间存在一对一的对应关系。我们的结果可以理解为Bonk，Heinonen和Koskela在“统一化Gromov双曲空间”中的主要结果的无限对应物，Astérisque270（2001）。
Uniformization theory of Gromov hypebolic spaces investigated by Bonk, Heinonen and Koskela, generalizes the case where a classical Poincaré ball type model is used as the starting point. In this paper, we develop this approach in the case where the underlying domain is unbounded, corresponding to the classical Poincaré half-space model. More precisely, we study conformal densities via Busemann functions on Gromov hyperbolic spaces and prove that the deformed spaces are unbounded uniform spaces. Furthermore, we show that there is a one-to-one correspondence between the bilipschitz classes of proper geodesic Gromov hyperbolic spaces that are roughly starlike with respect to a point on Gromov boundary and the quasisimilarity classes of unbounded locally compact uniform spaces. Our result can be understood as an unbounded counterpart of the main result of Bonk, Heinonen, and Koskela in "Uniformizing Gromov Hyperbolic Spaces", Astérisque 270 (2001).
