均匀化、$\partial$ - biLipschitz映射、球化和反演
Uniformization, $\partial$-biLipschitz maps, sphericalization, and inversion
摘要：
定义了一致度量空间之间的$\partial$ -biLipschitz同胚，并证明了这些映射总是拟莫比乌斯的.我们还证明了一个同胚是$\partial$ -biLipschitz是等价的映射biLipschitz在拟双曲度量在这些空间。这些主张的证明要求我们一致化拟双曲度量。我们进一步证明了Gromov双曲空间的所有容许一致化通过恒等映射彼此是拟莫比乌斯的，包括那些基于Gromov边界上一点的一致化。利用主要结果，我们然后表明，球化和反演操作是兼容的双曲空间在自然意义上的一致化。
英文摘要：
We define $\partial$-biLipschitz homeomorphisms between uniform metric spaces and show that these maps are always quasimöbius. We also show that a homeomorphism being $\partial$-biLipschitz is equivalent to the map biLipschitz in the quasihyperbolic metrics on these spaces. The proofs of these claims require us to uniformize the quasihyperbolic metric. We further show that all admissible uniformizations of a Gromov hyperbolic space are quasimöbius to one another by the identity map, including those uniformizations that are based at a point of the Gromov boundary. Using the main results we then show that the sphericalization and inversion operations are compatible with uniformization of hyperbolic spaces in a natural sense.
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