Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Tee, Ming Hong
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7216014357