论文标题
用Busemann功能统一的Gromov双曲线空间
Uniformizing Gromov hyperbolic spaces with Busemann functions
论文作者
论文摘要
给定一个完整的gromov双波利空间$ x $，它与gromov boundary $ \ partial_ {g} x $的点$ω$大致类似，我们使用基于$ω$的busemann函数来构建一个不完整的未结合的均匀统一的均匀均匀的均匀度量$ x _ {用$ x $的Gromov Boundare $ \partial_Ωx$相对于$ω$确定。这种统一的结构概括了从双曲机平面获得欧几里得上半平面的过程。此外，对于任意的度量空间$ z $，我们显示的是，有一个双曲线填充$ x $的$ z $，可以以这样的方式均匀化，以使边界$ \ partial x _ {\ varepsilon} $具有bilipschitz的bilipschitz识别，带有完整的$ \ \ bar {z} $ $ z $的$ \ bar {z} $。我们还证明，在CAT $（-1）$空间的情况下，可以在通常最佳的指数上完成此统一过程。
Given a complete Gromov hyperbolic space $X$ that is roughly starlike from a point $ω$ in its Gromov boundary $\partial_{G}X$, we use a Busemann function based at $ω$ to construct an incomplete unbounded uniform metric space $X_{\varepsilon}$ whose boundary $\partial X_{\varepsilon}$ can be canonically identified with the Gromov boundary $\partial_ωX$ of $X$ relative to $ω$. This uniformization construction generalizes the procedure used to obtain the Euclidean upper half plane from the hyperbolic plane. Furthermore we show, for an arbitrary metric space $Z$, that there is a hyperbolic filling $X$ of $Z$ that can be uniformized in such a way that the boundary $\partial X_{\varepsilon}$ has a biLipschitz identification with the completion $\bar{Z}$ of $Z$. We also prove that this uniformization procedure can be done at an exponent that is often optimal in the case of CAT$(-1)$ spaces.
