Local integrability of measurable conformal structures

Assume the Axiom of Choice. It implies Countable Choice (AC implies DC implies countable choice). Let $\mathrm{\Omega }\subseteq \mathbb{C}$ be a complex domain and let $\mu$ be a Beltrami coefficient on $\mathrm{\Omega }$ with $\mathrm{\parallel }\mu {\mathrm{\parallel }}_{\mathrm{\infty }}\le k$ for some $0\le k<1$ (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), A complex domain is a nonempty connected open subset of $\mathbb{C}$, Measurable Beltrami coefficients and measurable conformal structures).

(i) Local coordinates. For every $p\in \mathrm{\Omega }$, there are $r>0$ with $\overline{D(p,r)}\subseteq \mathrm{\Omega }$ and a complex domain $V\subseteq \mathbb{C}$ (A complex domain is a nonempty connected open subset of $\mathbb{C}$) with an orientation-preserving analytically quasiconformal homeomorphism $w:D(p,r)\to V$ (The ACL and Sobolev analytic definition of quasiconformality, Orientation-preserving homeomorphisms and the geometric definition of quasiconformality) whose weak derivatives satisfy $w_{\bar{z}}=\mu w_z$ almost everywhere (Weak solutions of the Beltrami equation) and whose Beltrami coefficient equals $\mu$ almost everywhere (The Beltrami coefficient and the maximal dilatation). Thus the measurable conformal structure defined by $\mu$ is locally equivalent to the standard one.

(ii) Uniqueness up to conformal maps. If $U,V_1,V_2\subseteq \mathbb{C}$ are complex domains, $U\subseteq \mathrm{\Omega }$, and $w_i:U\to V_i$ $(i=1,2)$ are orientation-preserving analytically quasiconformal homeomorphisms that solve the same Beltrami equation and have ${\mu }_{w_1}={\mu }_{w_2}=\mu$ almost everywhere, then $w_2\circ w_1^{-1}:V_1\to V_2$ is conformal, hence biholomorphic (Composition and inversion of quasiconformal maps and their Beltrami coefficients, Every 1-quasiconformal homeomorphism is conformal, Biholomorphic maps between complex domains). The transition maps between quasiconformal coordinates therefore differ by conformal postcomposition.

Lyubich, Ch. 2 Theorem 14.1, printed pp. 195–196, states the semi-local integrability result. §§14.1–14.2, printed p. 196, were read in full: uniqueness follows because the quotient of two solutions has vanishing $\bar{\mathrm{\partial }}$ and Weyl's lemma makes it conformal; the global theorem yields the local one by zero extension. The item writes out the coefficient extension and finite-chart restriction. Bishop, Ch. 3 §2, printed p. 88, Theorem 2.11, was read in full but is context only: its printed $K=(k+1)\mathrm{/}(k-1)$ is negative for $0\le k<1$, and the proof invokes an unresolved “Theorem ??” for coefficient convergence.

The stable global theorem and earlier12 area/inverse-null/composition interfaces supply the exact assertions used in this proof. The local arguments above retain their own stated hypotheses; this reconciliation is separate from root mathematical decisions and full-run certification.