拟线上拟对称嵌入的弱渐近对称性

中文摘要 AI 辅助

我们研究了与平面拟线相关的拟对称映射中弱渐近对称性（$\mathrm{WAS}$）与渐近对称性（$\mathrm{AS}$）之间的关系。为此，我们引入了一个形式上更弱的条件，称为等距弱渐近对称性（$\mathrm{EWAS}$），并证明了对于从$\mathbb{R}$到$\mathbb{C}$的拟对称嵌入，条件$\mathrm{AS}$、$\mathrm{WAS}$和$\mathrm{EWAS}$三者等价。接着，我们确立了对于从任意拟线到$\mathbb{R}$的拟对称同胚，$\mathrm{WAS}$蕴含$\mathrm{AS}$。更一般地，若$h\colon\Gamma_1\to\Gamma_2$是拟线之间的拟对称同胚且$\Gamma_1$是渐近共形的，则$\mathrm{WAS}$蕴含$\mathrm{AS}$。当$\Gamma_2$是渐近共形且$h$一致连续时，存在形式上对偶的结论。在有界拟圆的紧致情形下，这些结果回答了Brania和Yang提出的$\mathrm{WAS}$-$\mathrm{AS}$问题。

英文摘要

We investigate the relationship between weak asymptotic symmetry ($\mathrm{WAS}$) and asymptotic symmetry ($\mathrm{AS}$) for quasisymmetric maps associated with planar quasilines. To this end, we introduce a formally weaker condition, called equidistant weak asymptotic symmetry ($\mathrm{EWAS}$), and prove that, for quasisymmetric embeddings of $\mathbb{R}$ into $\mathbb{C}$, the three conditions $\mathrm{AS}$, $\mathrm{WAS}$, and $\mathrm{EWAS}$ are equivalent. We then establish that $\mathrm{WAS}$ implies $\mathrm{AS}$ for quasisymmetric homeomorphisms from an arbitrary quasiline onto $\mathbb{R}$. More generally, if $h\colonΓ_1\toΓ_2$ is a quasisymmetric homeomorphism between quasilines and $Γ_1$ is asymptotically conformal, then $\mathrm{WAS}$ implies $\mathrm{AS}$. A formally dual statement holds when $Γ_2$ is asymptotically conformal, provided that $h$ is uniformly continuous. In the compact setting of bounded quasicircles, these results yield an answer to the $\mathrm{WAS}$-$\mathrm{AS}$ problem posed by Brania and Yang.

发表机构

• Waseda University（早稻田大学）
• Peking University（北京大学）

机构由 AI 辅助整理，请以论文原文为准。