发表机构
Department of Mathematics, Boston College（波士顿学院数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用局部拟反射改进 Ahlfors 估计，证明渐近共形曲线围成区域的双曲与拟双曲度量渐近相似，并借助 Epstein 曲面表明互补区域的 Epstein--Poincaré 曲面在曲线附近任意接近。
AI 中文摘要
Ahlfors--Weill 构造将 Schwarzian 范数小于 1/2 的单叶函数 $f$ 显式延拓为黎曼球面上的拟共形同胚，这一延拓可通过 Epstein 曲面从几何上加以理解。对于渐近共形曲线，均匀化映射可延拓到更大的圆盘，且 Loewner 理论可用于验证该延拓是拟共形的。在本文中，我们构造了一个具有此类延拓的局部拟反射。利用该局部拟反射，我们改进了 Ahlfors 的经典估计，并证明由渐近共形曲线围成的区域具有渐近相似的双曲度量与拟双曲度量。此外，通过将这些估计与经 Epstein 曲面对该延拓的几何重新解释相结合，我们证明了互补区域的 Epstein--Poincaré 曲面在曲线附近彼此任意接近。
英文摘要
The Ahlfors--Weill construction explicitly extends a univalent function $f$ with Schwarzian norm less than $1/2$ to a quasiconformal homeomorphism of the Riemann sphere, an extension that can be understood geometrically through Epstein surfaces. For asymptotically conformal curves, the uniformization map can be extended to a larger disk, and Loewner theory can be used to verify that this extension is quasiconformal. In this paper, we construct a local quasireflection with such an extension. Using this local quasireflection, we modify classical estimates of Ahlfors and show that domains bounded by asymptotically conformal curves have asymptotically similar hyperbolic and quasihyperbolic metrics. Furthermore, by combining these estimates with a geometric reinterpretation of the extension through Epstein surfaces, we show that the Epstein-Poincaré surfaces of the complementary domains are arbitrarily close to one another near the curve.
