Measurable Beltrami coefficients and measurable conformal structures

Assume Countable Choice and identify the complex plane with the Euclidean plane by $z=x+iy\leftrightarrow (x,y)$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves). All planar domains carry two-dimensional Lebesgue area measure.

(a) Plane domains. Let $\mathrm{\Omega }\subseteq \mathbb{C}$ be a complex domain (A complex domain is a nonempty connected open subset of $\mathbb{C}$). A Beltrami coefficient on $\mathrm{\Omega }$ is an almost-everywhere class of Lebesgue-measurable functions $\mu :\mathrm{\Omega }\to \mathbb{C}$ with finite essential supremum $\mathrm{\parallel }\mu {\mathrm{\parallel }}_{\mathrm{\infty }}$ (Borel measurable and Lebesgue measurable functions on ${\mathbb{R}}^n$, The space $L^{\mathrm{\infty }}(\mu )$ of essentially bounded measurable functions). Here $L^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{C})$ means this class with norm ${\mathrm{ess\hspace{0.17em}sup}}_{z\in \mathrm{\Omega }}\mathrm{\mid }\mu (z)\mathrm{\mid }$; equivalently, its real and imaginary coordinate functions belong to the real-valued $L^{\mathrm{\infty }}(\mathrm{\Omega })$. The defining bound is strict: $\mathrm{\parallel }\mu {\mathrm{\parallel }}_{\mathrm{\infty }}<1.$ Thus $\mathrm{\mid }\mu \mathrm{\mid }<1$ almost everywhere, and representatives differing on a Lebesgue-null set determine the same coefficient. Its dilatation is $K(\mu ):=\frac{1+\mathrm{\parallel }\mu {\mathrm{\parallel }}_{\mathrm{\infty }}}{1-\mathrm{\parallel }\mu {\mathrm{\parallel }}_{\mathrm{\infty }}}\in [1,\mathrm{\infty })\mathrm{.}$ In particular, an essentially bounded measurable function with $\mathrm{\parallel }\mu {\mathrm{\parallel }}_{\mathrm{\infty }}=1$ is not a Beltrami coefficient.

(b) Ellipse-field reading. Regard an ellipse as a shape, ignoring positive rescaling. A measurable field of ellipses of bounded eccentricity has, almost everywhere, measurable major and minor semiaxes $a(z)\ge b(z)>0$ and a measurable unoriented major-axis direction $\theta (z)(\mathrm{m}\mathrm{o}\mathrm{d}\pi )$, with $a(z)\mathrm{/}b(z)\le C$ for some finite constant $C$. Such a field determines the coefficient $\mu (z)=\frac{a(z)-b(z)}{a(z)+b(z)}{e}^{2i\theta (z)}\mathrm{.}$ When $a(z)=b(z)$ the ellipse is a circle and this formula gives $\mu (z)=0$, with no distinguished direction. Conversely, at every Lebesgue point of a representative of $\mu$, its ellipse has major-to-minor semiaxis ratio $(1+\mathrm{\mid }\mu (z)\mathrm{\mid })\mathrm{/}(1-\mathrm{\mid }\mu (z)\mathrm{\mid })$ and, when $\mu (z)\mathrm{\ne }0$, major-axis direction $\frac{1}{2}\arg \mu (z)(\mathrm{m}\mathrm{o}\mathrm{d}\pi )$. These formulas identify measurable coefficients with measurable conformal structures up to null sets, and $K(\mu )={\mathrm{ess\hspace{0.17em}sup}}_{z\in \mathrm{\Omega }}\frac{1+\mathrm{\mid }\mu (z)\mathrm{\mid }}{1-\mathrm{\mid }\mu (z)\mathrm{\mid }}\mathrm{.}$

(c) Biholomorphic change of coordinates. If $\psi :{\mathrm{\Omega }}^{\mathrm{\prime }}\to \mathrm{\Omega }$ is biholomorphic (Biholomorphic maps between complex domains) and $\mu$ is a Beltrami coefficient on $\mathrm{\Omega }$, define its pullback by $({\psi }^{\ast }\mu )(\zeta ):=\mu (\psi (\zeta ))\frac{\overline{{\psi }^{\mathrm{\prime }}(\zeta )}}{{\psi }^{\mathrm{\prime }}(\zeta )},\zeta \in {\mathrm{\Omega }}^{\mathrm{\prime }}\mathrm{.}$ Since $\psi$ and ${\psi }^{-1}$ are holomorphic, they are $C^1$ in real coordinates (Holomorphic functions are real analytic and smooth in their two real coordinates); together with their inverse identities this makes $\psi$ a $C^1$ diffeomorphism. The complex differentiability criterion identifies the real derivative $D\psi (\zeta )$ with multiplication by ${\psi }^{\mathrm{\prime }}(\zeta )$ (Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with ${\mathrm{\partial }}_{\bar{z}}f=0$, or with the Cauchy–Riemann equations). The real chain rule applied to ${\psi }^{-1}\circ \psi =\mathrm{id}$ makes this derivative invertible, hence ${\psi }^{\mathrm{\prime }}(\zeta )\mathrm{\ne }0$. Therefore the factor has modulus one (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive). Both $\psi$ and ${\psi }^{-1}$ send Lebesgue-null sets to null sets, so composition preserves Lebesgue measurability and essential supremum (A C^1 diffeomorphism maps Lebesgue null sets to Lebesgue null sets). Consequently ${\psi }^{\ast }\mu$ is a Beltrami coefficient, $\mathrm{\parallel }{\psi }^{\ast }\mu {\mathrm{\parallel }}_{\mathrm{\infty }}=\mathrm{\parallel }\mu {\mathrm{\parallel }}_{\mathrm{\infty }}$, and $K({\psi }^{\ast }\mu )=K(\mu )$. The chain rule gives functoriality: for biholomorphisms $\chi :{\mathrm{\Omega }}^{\mathrm{\prime }\mathrm{\prime }}\to {\mathrm{\Omega }}^{\mathrm{\prime }}$ and $\psi :{\mathrm{\Omega }}^{\mathrm{\prime }}\to \mathrm{\Omega }$, $(\psi \circ \chi {)}^{\ast }\mu ={\chi }^{\ast }({\psi }^{\ast }\mu )\mathrm{.}$

(d) The Riemann sphere. Write $z$ for the finite chart and $w=1\mathrm{/}z$ for the chart at infinity (The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity, The Riemann sphere is the published one-point compactification of the complex plane). A Beltrami coefficient on $\widehat{\mathbb{C}}$ is an almost-everywhere class of measurable chart representatives related on their overlap by (c). The overlap transition is biholomorphic and preserves null sets by the cited $C^1$ null-set lemma, so the chartwise almost-everywhere notion is consistent; no common global scalar representative is intended. Equivalently, a coefficient on the sphere is determined by a coefficient ${\mu }_0$ on the finite chart $\mathbb{C}$ with $\mathrm{\parallel }{\mu }_0{\mathrm{\parallel }}_{\mathrm{\infty }}<1$; its expression in the infinity chart is ${\mu }_{\mathrm{\infty }}(w)={\mu }_0(1\mathrm{/}w)\frac{w^2}{{\bar{w}}^2},w\mathrm{\ne }0,$ and the value at $w=0$ is immaterial to the almost-everywhere class. This is the pullback law (c) for $\psi (w)=1\mathrm{/}w$, since $\overline{{\psi }^{\mathrm{\prime }}(w)}\mathrm{/}{\psi }^{\mathrm{\prime }}(w)=w^2\mathrm{/}{\bar{w}}^2$; the single missing point is null. In particular, the condition $\mathrm{\parallel }\mu {\mathrm{\parallel }}_{\mathrm{\infty }}<1$ and the dilatation $K(\mu )$ are independent of the chosen chart expression.