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Uniformization of cofat domains on metric two-spheres

Chengxi Li, Kai Rajala

TL;DR

This work extends Schramm's cofat uniformization to cofat domains on upper Ahlfors $2$-regular metric two-spheres, proving the existence of a $ rac{\pi}{2}$-quasiconformal map to a circle domain that preserves both point- and nontrivial components. The authors develop a transboundary modulus framework and an approximation scheme by finitely connected domains, combining Rajala's uniformization with circle domain uniformization to obtain a limit map $f$ with controlled boundary behavior. They establish a uniform modulus bound crucial for the convergence analysis, and prove that the circle-domain image preserves the component structure, while also constructing sharp counterexamples showing the necessity of the cofat condition and the sharpness of the exponent in $\,\ell^{\alpha}$-type decay. The results connect non-smooth uniformization theory with classical Koebe-type questions, providing both positive uniformization results and sharp obstructions in generality.

Abstract

We extend \emph{Schramm's cofat uniformization theorem} to cofat domains on upper Ahlfors 2-regular metric two-spheres $X$. Specifically, we show that if $Ω\subset X$ is a cofat domain, then there exists a $\fracπ{2}$-quasiconformal homeomorphism $f: Ω\to D$ onto a circle domain $D \subset \mathbb{S}^2$. Moreover, $f$ preserves the point-components and non-trivial complementary components. We also construct examples which show that the above conclusions are not true for countably connected $\ell^α$-subdomains of $\mathbb{S}^2$.

Uniformization of cofat domains on metric two-spheres

TL;DR

This work extends Schramm's cofat uniformization to cofat domains on upper Ahlfors -regular metric two-spheres, proving the existence of a -quasiconformal map to a circle domain that preserves both point- and nontrivial components. The authors develop a transboundary modulus framework and an approximation scheme by finitely connected domains, combining Rajala's uniformization with circle domain uniformization to obtain a limit map with controlled boundary behavior. They establish a uniform modulus bound crucial for the convergence analysis, and prove that the circle-domain image preserves the component structure, while also constructing sharp counterexamples showing the necessity of the cofat condition and the sharpness of the exponent in -type decay. The results connect non-smooth uniformization theory with classical Koebe-type questions, providing both positive uniformization results and sharp obstructions in generality.

Abstract

We extend \emph{Schramm's cofat uniformization theorem} to cofat domains on upper Ahlfors 2-regular metric two-spheres . Specifically, we show that if is a cofat domain, then there exists a -quasiconformal homeomorphism onto a circle domain . Moreover, preserves the point-components and non-trivial complementary components. We also construct examples which show that the above conclusions are not true for countably connected -subdomains of .

Paper Structure

This paper contains 10 sections, 21 theorems, 95 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conformal Modulus
  4. Transboundary Modulus
  5. Approximation with finitely connected domains
  6. Modulus estimates
  7. Proof of Proposition \ref{['unibdd']}
  8. Proof of Theorem \ref{['bz']}
  9. Proof of Theorem \ref{['nontop2']}
  10. Proof of Theorem \ref{['ptonon2']}

Key Result

Theorem 1.1

Suppose that $X$ is an upper Ahlfors $2$-regular metric two-sphere. If $\Omega \subset X$ is a cofat domain, then there exists a $\frac{\pi}{2}$-quasiconformal homeomorphism $f\colon\Omega \to D$ onto a circle domain $D \subset \mathbb{S}^2$. Moreover,

Figures (4)

  • Figure 1: Part of a sample curve $\gamma=\gamma_1 * p^1 * \cdots * p^n$ that satisfies $\gamma(t) \notin \mathscr{G}$, each $\operatorname{diam}(p^i) < \frac{r}{10}$.
  • Figure 2: Some complementary components of $U_\alpha$ and the Jordan curve $J \subset U_\alpha$.
  • Figure 3: The annuli $A_{1,0}$ and $A_{2,0}$ in the definition of $\rho$.
  • Figure 4: Some complementary components of $\Omega_\alpha$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Conformal modulus
  • Definition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Definition 2.5
  • Theorem 2.6
  • ...and 27 more