论文标题
平面域中的双曲线距离与quasihyperbolic距离
Hyperbolic Distance versus Quasihyperbolic Distance in Plane Domains
论文作者
论文摘要
我们检查欧几里得平面域具有其双曲线或准透明距离。我们证明相关的度量空间在且仅当它们为Bi-Lipschitz等效时是准对称等效的。另一方面，对于Gromov双曲线域，两个相应的Gromov边界总是在准对称上等效的。令人惊讶的是，对于任何有限连接的双曲线结构域，这两个度量空间始终是准对象的。我们构建一个示例，其中空间不是准对象等效的。
We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for Gromov hyperbolic domains, the two corresponding Gromov boundaries are always quasisymmetrically equivalent. Surprisingly, for any finitely connected hyperbolic domain, these two metric spaces are always quasiisometrically equivalent. We construct an example where the spaces are not quasiisometrically equivalent.
