Ahlfors--Weill Extension, Asymptotically Conformal Curves and Epstein--Poincaré Surfaces

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.

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BibTeX RIS

BibTeX

@misc{tee2026ahlforsweill,
  title = {{Ahlfors--Weill Extension, Asymptotically Conformal Curves and Epstein--Poincaré Surfaces}},
  author = {Ming Hong Tee},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02171},
  eprint = {2610.02171},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02171},
}

RIS

TY  - UNPB
TI  - Ahlfors--Weill Extension, Asymptotically Conformal Curves and Epstein--Poincaré Surfaces
AU  - Ming Hong Tee
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.02171
UR  - https://arxiv.org/abs/2610.02171
AB  - 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.
ER  -

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