Extremal Length and Planar Quasiconformality

• Mikhail Lyubich, Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I, Ch. 2 §§11.3–11.5. Section 11.3 defines distributional partial derivatives and gives the ACL criterion for homeomorphisms of planar domains; §11.4 defines quasiconformality by local integrability of these derivatives and bounded dilatation; Proposition 2.11 proves local square-integrability for quasiconformal maps.
• Christopher J. Bishop, Quasiconformal Mappings, Ch. 2 §1 and Ch. 3 §4, equation (4.1). The example there uses the Cantor singular function to show that the differential inequality alone does not imply ACL. Its printed derivative sentence has a typo: for $f(x+iy)=x+g(x)+iy$, one has ${\mathrm{\partial }}_{z}f=1$ and ${\mathrm{\partial }}_{\bar{z}}f=0$ almost everywhere; these are the values used below.

Assume the Axiom of Choice (The Axiom of Choice). Let $\mathrm{\Omega },{\mathrm{\Omega }}^{\mathrm{\prime }}\subseteq \mathbb{C}$ be complex domains (A complex domain is a nonempty connected open subset of $\mathbb{C}$) and $f:\mathrm{\Omega }\to {\mathrm{\Omega }}^{\mathrm{\prime }}$ a homeomorphism (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological). Read $f$ as a map into ${\mathbb{R}}^2$; continuity makes its components locally square integrable. Recall that $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,2}(\mathrm{\Omega })$ (Integer-order Sobolev spaces and their norms) means that the components of $f$ admit weak first partial derivatives (Weak derivative of a locally integrable function) in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$; equivalently, their almost-everywhere classes have ACL representatives in both coordinate directions whose measurable classical coordinate derivatives belong to $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$ (Absolute continuity on almost every coordinate line, The ACL characterisation of $W^{1,p}$). ACL alone does not assert this derivative integrability. Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) is included in AC for the completed-product Fubini interfaces. On almost every coordinate line the continuous map $f$ agrees almost everywhere with its ACL representative, hence everywhere by continuity of both restrictions; its classical line derivatives then represent its weak derivatives.

Let $K\ge 1$ and put $k=(K-1)\mathrm{/}(K+1)\in [0,1)$. The homeomorphism $f$ is $K$-quasiconformal in the analytic sense when

(A1) $f\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,2}(\mathrm{\Omega })$; and

(A2) its weak Wirtinger derivatives ${\mathrm{\partial }}_{z}f,{\mathrm{\partial }}_{\bar{z}}f$ (The Wirtinger derivatives ${\mathrm{\partial }}_{z}f$ and ${\mathrm{\partial }}_{\bar{z}}f$, and antiholomorphic functions) lie in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(\mathrm{\Omega })$ and satisfy $\mathrm{\mid }{\mathrm{\partial }}_{\bar{z}}f(z)\mathrm{\mid }\le k\mathrm{\mid }{\mathrm{\partial }}_{z}f(z)\mathrm{\mid }\text{for almost every }z\in \mathrm{\Omega }\mathrm{.}$

The inequality is a statement about $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$ classes: it is independent of the choice of representative of $f$ and of the Borel representatives of its weak Wirtinger derivatives, since each such representative agrees almost everywhere with its class (Weak differentiation ignores null-set changes). The map is analytically quasiconformal if it is $K$-analytically quasiconformal for some finite $K$. Its minimal constant is recovered from the Beltrami coefficient defined later on this page.

Orientation and the regularity requirement. Inequality (A2) gives $J_f=\mathrm{\mid }{\mathrm{\partial }}_{z}f{\mathrm{\mid }}^2-\mathrm{\mid }{\mathrm{\partial }}_{\bar{z}}f{\mathrm{\mid }}^2\ge (1-k^2)\mathrm{\mid }{\mathrm{\partial }}_{z}f{\mathrm{\mid }}^2\ge 0$ almost everywhere, with respect to planar Lebesgue measure (The nonnegative Lebesgue integral). A classical derivative inequality alone does not replace (A1): on $(0,1{)}^2$, the map $f(x+iy)=x+g(x)+iy$, where $g$ is the Cantor function, is a homeomorphism onto $(0,2)\times (0,1)$ (The map $x\mapsto x+c(x)$ is a homeomorphism from $[0,1]$ onto $[0,2]$). The Cantor function is continuous, nonconstant, and locally constant off the null Cantor set, so $g^{\mathrm{\prime }}=0$ almost everywhere (The Cantor function is well defined, satisfies $c(x)\le c(y)$ whenever $x\le y$, is surjective onto $[0,1]$, and is constant on every interval removed from the Cantor set, The Cantor set is an uncountable subset of $\mathbb{R}$ of Lebesgue measure zero). Thus the classical Wirtinger derivatives are ${\mathrm{\partial }}_{z}f=1$ and ${\mathrm{\partial }}_{\bar{z}}f=0$ almost everywhere. But $g$ cannot be absolutely continuous on every compact subinterval of $(0,1)$: the fundamental theorem would make it constant there, and then continuity would contradict $g(0)=0$, $g(1)=1$ (Fundamental theorem of calculus for absolutely continuous functions). Subtracting the identity shows that $f$ fails ACL on every horizontal line and therefore fails (A1). Its classical inequality is not the weak-derivative assertion (A2). Orientation preservation follows from (A1) together with (A2), as proved by the equivalence theorem on this page.

• Christopher J. Bishop, Quasiconformal Mappings, Ch. 1 §1, printed pp. 1–3. Bishop defines admissibility by ${\mathrm{\ell }}_{\rho }(\mathrm{\Gamma })\ge 1$, modulus by the infimum of $\int {\rho }^2$, and extremal length as the reciprocal modulus; he also states that the density may be taken Borel.
• Mikhail Lyubich, Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I, Ch. 1 §6.1, printed pp. 119–120. Lyubich defines $L_{\rho }(\mathrm{\Gamma })={\mathrm{\ell }}_{\rho }(\mathrm{\Gamma }{)}^2\mathrm{/}{m}_{\rho }(U)$ and takes its supremum over finite-mass metrics; the reciprocal is extremal width and is also the infimum over metrics whose length on every curve is at least $1$.
• Lars Ahlfors and Arne Beurling, Conformal Invariants and Function-Theoretic Null-Sets, §4, printed pp. 114–115. Their supremum convention for extremal length agrees with $\lambda$ below. Their Lemmas 4–5 give the rectangle and round-annulus constants after the curve family is specified.

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Fix a complex domain $\mathrm{\Omega }\subseteq \mathbb{C}$ (A complex domain is a nonempty connected open subset of $\mathbb{C}$), read as an open subset of the Euclidean plane, with its Borel $\sigma$-algebra (The Borel sigma-algebra of a topological space) and planar Lebesgue area measure (Lebesgue measure is a Radon measure on R^n).

A path in $\mathrm{\Omega }$ is a continuous map $\gamma :[a,b]\to \mathrm{\Omega }$. It is rectifiable when its two coordinate functions have bounded variation (A path in ${\mathbb{R}}^n$ is rectifiable exactly when every coordinate has bounded variation), and $s_{\gamma }$ denotes its arc-length function (The arc-length function $s_{\gamma }(t)=L(\gamma {\mathrm{\mid }}_{[a,t]})$ of a rectifiable path). A path family $\mathrm{\Gamma }$ is a set of paths. When curves joining specified boundary sets of $\mathrm{\Omega }$ are used, allow paths $\gamma :[a,b]\to \bar{\mathrm{\Omega }}$ with $\gamma ((a,b))\subseteq \mathrm{\Omega }$ and endpoints in the named sets; read complex paths as planar rectifiable paths (Complex contours as planar rectifiable paths: the Euclidean, coordinate-BV, and piecewise-C1 dictionaries). Only the interior contributes to length, since each density below is extended by $0$ outside $\mathrm{\Omega }$.

Let $\rho :\mathrm{\Omega }\to [0,+\mathrm{\infty }]$ be Borel measurable with respect to the Borel $\sigma$-algebra of $\mathrm{\Omega }$ (Extended-real-valued measurable functions). Extending it by $0$ on $\mathbb{C}\setminus \mathrm{\Omega }$ gives a Borel function on the plane by the trace identity for subspaces (The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra). For a rectifiable path $\gamma :[a,b]\to \bar{\mathrm{\Omega }}$, its arc-length function is continuous and nondecreasing (The arc-length function is continuous and nondecreasing, with increments equal to subpath lengths; it is strictly increasing exactly when no nondegenerate subpath is constant). Extend $s_{\gamma }$ to a continuous nondecreasing function $S_{\gamma }:\mathbb{R}\to \mathbb{R}$ by $S_{\gamma }(t)=0$ for $t\le a$, $S_{\gamma }(t)=s_{\gamma }(t)$ for $a\le t\le b$, and $S_{\gamma }(t)=s_{\gamma }(b)$ for $t\ge b$. Countable Choice gives the associated Lebesgue-Stieltjes Borel measure $d{S}_{\gamma }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), Assuming countable choice, a nondecreasing right-continuous function defines a Borel measure on $\mathbb{R}$). Define the $\rho$-length by ${\mathrm{\ell }}_{\rho }(\gamma ):={\int }_{[a,b]}\rho (\gamma (t))d{S}_{\gamma }(t),$ using the nonnegative Lebesgue integral (The nonnegative Lebesgue integral). This value lies in $[0,+\mathrm{\infty }]$; set ${\mathrm{\ell }}_{\rho }(\gamma )=+\mathrm{\infty }$ when $\gamma$ is not rectifiable. For finite-valued continuous $\rho$ and a rectifiable path whose full trace lies in $\mathrm{\Omega }$, this agrees with the published absolute line integral ${\int }_{\gamma }\rho \mathrm{\mid }dz\mathrm{\mid }$ (The absolute line integral over a rectifiable path using its arc-length function, For a continuous integrand, the Riemann-Stieltjes and Lebesgue-Stieltjes integrals agree, Continuous integrands have complex and absolute line integrals along every rectifiable path). Continuity only on $\mathrm{\Omega }$ does not assert that the zero extension is continuous at boundary endpoints. The continuity of $S_{\gamma }$ makes $d{S}_{\gamma }$ atomless, so changing the integrand on finitely many parameter values does not change the length.

For a path family put ${\mathrm{\ell }}_{\rho }(\mathrm{\Gamma }):={\inf }_{\gamma \in \mathrm{\Gamma }}{\mathrm{\ell }}_{\rho }(\gamma )$, with $\inf \mathrm{\varnothing }:=+\mathrm{\infty }$, and define its area by $A(\rho ):={\int }_{\mathrm{\Omega }}{\rho }^{2}dA\in [0,+\mathrm{\infty }]\mathrm{.}$

The extremal length of $\mathrm{\Gamma }$ is $\lambda (\mathrm{\Gamma }):={\sup }_{\rho }\frac{{\mathrm{\ell }}_{\rho }(\mathrm{\Gamma }{)}^2}{A(\rho )}\in [0,+\mathrm{\infty }],$ where the supremum is over Borel $\rho :\mathrm{\Omega }\to [0,+\mathrm{\infty }]$ with $0<A(\rho )<+\mathrm{\infty }$. A supremum of an empty set is $0$; since $\mathrm{\Omega }$ is a nonempty open domain, it contains a nondegenerate rectangle $Q$, and its indicator has positive finite area (A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included). The value ${\mathrm{\ell }}_{\rho }(\mathrm{\Gamma })=+\mathrm{\infty }$ is allowed. The curve-family modulus is the reciprocal $\mu (\mathrm{\Gamma }):=\frac{1}{\lambda (\mathrm{\Gamma })}\in [0,+\mathrm{\infty }],$ with $1\mathrm{/}0:=+\mathrm{\infty }$ and $1\mathrm{/}+\mathrm{\infty }:=0$. Thus $\lambda (\mathrm{\varnothing })=+\mathrm{\infty }$ and $\mu (\mathrm{\varnothing })=0$, while any family containing a constant path has $\lambda (\mathrm{\Gamma })=0$ and $\mu (\mathrm{\Gamma })=+\mathrm{\infty }$.

Conventions. (1) This library defines extremal length by the displayed supremum and calls its reciprocal the modulus. Sources that define modulus by ${\inf }_{\rho }\int {\rho }^2$ over metrics with ${\mathrm{\ell }}_{\rho }(\mathrm{\Gamma })\ge 1$ call $\mu (\mathrm{\Gamma })$ the modulus and $\lambda (\mathrm{\Gamma })$ the extremal length. (2) Extending $\rho$ by zero outside $\mathrm{\Omega }$ makes densities supported in $\mathrm{\Omega }$ admissible and makes the value independent of an ambient enlargement; parameterization invariance, this ambient-domain independence, and agreement with the continuous line integral are the well-definedness obligations recorded for The rho-length and the extremal length are well defined ↗. (3) The page's rectangle and round-annulus constants are fixed by the computations stated there, and each cited source is translated into this library convention.

• Christopher J. Bishop, Quasiconformal Mappings, Ch. 2 §1, printed pp. 49–51. For an orientation-preserving nonsingular real-linear map, Bishop obtains the complex dilatation $\mu =f_{\bar{z}}\mathrm{/}{f}_z$, $\mathrm{\mid }\mu \mathrm{\mid }<1$, and the dilatation $D=(1+\mathrm{\mid }\mu \mathrm{\mid })\mathrm{/}(1-\mathrm{\mid }\mu \mathrm{\mid })$, equivalently $\mathrm{\mid }\mu \mathrm{\mid }=(D-1)\mathrm{/}(D+1)$.
• Mikhail Lyubich, Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I, Ch. 2 §§11.1.2–11.3, printed pp. 177–181. Section 11.1.2 defines the pointwise Beltrami coefficient; equation (11.3) on p. 178 gives $\mathrm{Dil}=(1+\mathrm{\mid }\mu \mathrm{\mid })\mathrm{/}(1-\mathrm{\mid }\mu \mathrm{\mid })$; §11.3 adds the ACL/distributional regularity and bounded-dilatation requirements. The last K-conversion on p. 181 prints denominator $k-1$; this is a sign typo, so the correct conversion from Bishop and the library definition is used here.

Assume the Axiom of Choice. Let $f:\mathrm{\Omega }\to {\mathrm{\Omega }}^{\mathrm{\prime }}$ be a homeomorphism of complex domains whose components lie in $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,2}(\mathrm{\Omega })$, with weak Wirtinger derivative classes as in The ACL and Sobolev analytic definition of quasiconformality, The Wirtinger derivatives ${\mathrm{\partial }}_{z}f$ and ${\mathrm{\partial }}_{\bar{z}}f$, and antiholomorphic functions, and Weak derivative of a locally integrable function.

Choose finite Borel representatives of the real and imaginary components of these derivative classes. Such representatives exist: extend each component by zero outside $\mathrm{\Omega }$, use that Lebesgue measure is the completion of its Borel restriction and that every completion-measurable function equals a Borel function almost everywhere (Borel measurable and Lebesgue measurable functions on ${\mathbb{R}}^n$, $\mathcal{L}({\mathbb{R}}^n)$ is exactly the completion of the restriction of ${\lambda }_n$ to the Borel sets, A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra), then restrict the resulting Borel representatives to $\mathrm{\Omega }$ using the Borel trace identity (The Borel sigma-algebra of a topological space, The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra). Replacing infinite values on the resulting Borel null sets by zero gives finite representatives.

The Beltrami coefficient of $f$ is the Borel function ${\mu }_f(z):=\{\begin{array}{ll}{\mathrm{\partial }}_{\bar{z}}f(z)\mathrm{/}{\mathrm{\partial }}_{z}f(z),& {\mathrm{\partial }}_{z}f(z)\mathrm{\ne }0,\\ 0,& {\mathrm{\partial }}_{z}f(z)=0.\end{array}$ It is a finite-valued measurable function modulo equality almost everywhere. Changing the chosen Borel representatives changes ${\mu }_f$ only on a Lebesgue-null set, so this class is independent of those choices. For a general $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,2}$ homeomorphism, ${\mu }_f$ need not be essentially bounded. If $f$ is $K$-analytically quasiconformal and $k=(K-1)\mathrm{/}(K+1)$, then $\mathrm{\mid }{\mu }_f\mathrm{\mid }\le k<1$ almost everywhere and it defines a complex $L^{\mathrm{\infty }}(\mathrm{\Omega })$ class (Complex Lp classes and Euclidean test-function conventions). The value zero on $\{{\mathrm{\partial }}_{z}f=0\}$ is a fixed convention; analytic quasiconformality gives ${\mathrm{\partial }}_{\bar{z}}f=0$ almost everywhere on that set.

Let $m_f=\mathrm{\parallel }{\mu }_f{\mathrm{\parallel }}_{\mathrm{\infty }}$ be the essential supremum of $\mathrm{\mid }{\mu }_f\mathrm{\mid }$ (The essential supremum of a measurable function with respect to a measure). Define the maximal dilatation by $K_f:=\{\begin{array}{ll}{\displaystyle \frac{1+m_f}{1-m_f}},& m_f<1\text{ and }{\mathrm{\partial }}_{\bar{z}}f=0\text{ a.e. on }\{{\mathrm{\partial }}_{z}f=0\},\\ +\mathrm{\infty },& \text{otherwise.}\end{array}$ Then $K_f\in [1,+\mathrm{\infty }]$, and for every finite $K\ge 1$, $f$ is analytically $K$-quasiconformal exactly when $K_f\le K$. Indeed, off $\{{\mathrm{\partial }}_{z}f=0\}$ the analytic inequality is equivalent to $\mathrm{\mid }{\mu }_f\mathrm{\mid }\le (K-1)\mathrm{/}(K+1)$, and on that set the separate derivative condition in the definition of $K_f$ is exactly what makes the inequality hold. For finite $m_f$, intersecting the almost-everywhere bounds $\mathrm{\mid }{\mu }_f\mathrm{\mid }\le m_f+1\mathrm{/}n$ gives $\mathrm{\mid }{\mu }_f\mathrm{\mid }\le m_f$ almost everywhere. For an analytically quasiconformal map, the separate derivative condition holds and $m_f<1$, so $K_f=(1+m_f)\mathrm{/}(1-m_f)$ is the least admissible constant. In this class, $K_f=1$ exactly when ${\mu }_f=0$ almost everywhere, equivalently when ${\mathrm{\partial }}_{\bar{z}}f=0$ as an $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2$ class. The separate one-quasiconformal theorem on this page supplies the holomorphic conclusion in that case; it is not an input to this definition. The nullity of $\{{\mathrm{\partial }}_{z}f=0\}$ will follow from the inverse theorem's area formula and null-set properties; it is not an assumption here.

• Christopher J. Bishop, Quasiconformal Mappings, Ch. 1 §1, printed pp. 1–8. The notes define curve length by integrating a nonnegative Borel density against arclength and observe that densities may be set to zero outside the domain.
• Mikhail Lyubich, Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I, Ch. 1 §6.1 and Exercise 6.2, printed pp. 119–120. The exercise records ambient-surface independence; the arguments below supply the measure-theoretic details for the library's finite-positive-area convention, including its zero-area edge case.

Assume Countable Choice. Let $\mathrm{\Omega }\subseteq {\mathrm{\Omega }}^{\mathrm{\prime }}$ be complex domains, let $\mathrm{\Gamma }$ be a path family in $\mathrm{\Omega }$, and let $\rho ,\sigma :\mathrm{\Omega }\to [0,+\mathrm{\infty }]$ be Borel with $\rho \le \sigma$. Let ${\mathrm{\ell }}_{\rho }$, $A$, $\lambda$, and $\mu$ be as in Extremal length and the curve-family modulus of a path family. Then:

(i) Parameterization independence. If $\gamma :[a,b]\to \mathrm{\Omega }$ is rectifiable and $h:[a^{\mathrm{\prime }},b^{\mathrm{\prime }}]\to [a,b]$ is continuous, nondecreasing, and onto, then ${\mathrm{\ell }}_{\rho }(\gamma \circ h)={\mathrm{\ell }}_{\rho }(\gamma )$. With the arc-length parametrization $\tilde{\gamma }:[0,L]\to \mathrm{\Omega }$ of Every rectifiable path factors through its arc-length function as a unit-speed path on $[0,L]$, one has ${\mathrm{\ell }}_{\rho }(\gamma )={\int }_0^L\rho (\tilde{\gamma }(s))ds\mathrm{.}$

(ii) Subpath additivity. Write ${\nu }_{\gamma }=d{S}_{\gamma }$ for the Lebesgue-Stieltjes measure of the extended arc-length function in Extremal length and the curve-family modulus of a path family. For $a\le u\le v\le b$, ${\mathrm{\ell }}_{\rho }(\gamma {\mathrm{\mid }}_{[u,v]})={\int }_{[u,v]}\rho (\gamma (t))d{\nu }_{\gamma }(t)\mathrm{.}$ Consequently, if $[u_j,v_j]\subseteq [a,b]$ are pairwise disjoint parameter intervals, then ${\sum }_j{\mathrm{\ell }}_{\rho }(\gamma {\mathrm{\mid }}_{[u_j,v_j]})\le {\mathrm{\ell }}_{\rho }(\gamma )$.

(iii) Agreement with the absolute line integral. If $\rho$ is finite-valued and continuous on $\mathrm{\Omega }$, then for every rectifiable path $\gamma$ in $\mathrm{\Omega }$ the value ${\mathrm{\ell }}_{\rho }(\gamma )$ equals the published absolute line integral ${\int }_{\gamma }\rho \mathrm{\mid }dz\mathrm{\mid }$ (The absolute line integral over a rectifiable path using its arc-length function, Continuous integrands have complex and absolute line integrals along every rectifiable path). In particular, ${\mathrm{\ell }}_1(\gamma )$ is the arc length of $\gamma$.

(iv) Monotonicity and area additivity. For every path, ${\mathrm{\ell }}_{\rho }(\gamma )\le {\mathrm{\ell }}_{\sigma }(\gamma )$, hence ${\mathrm{\ell }}_{\rho }(\mathrm{\Gamma })\le {\mathrm{\ell }}_{\sigma }(\mathrm{\Gamma })$. If pairwise disjoint Borel sets $E_1,\dots ,E_k\subseteq \mathrm{\Omega }$ satisfy $\rho =0$ off their union, then $A(\rho )={\sum }_{j=1}^k{\int }_{E_j}{\rho }^{2}dA\mathrm{.}$

(v) Independence of the ambient domain. Extending $\rho$ by zero to ${\mathrm{\Omega }}^{\mathrm{\prime }}$ does not change the $\rho$-length of any path in $\mathrm{\Omega }$ or its area. Consequently, $\lambda (\mathrm{\Gamma })$ and $\mu (\mathrm{\Gamma })$ computed using metrics on $\mathrm{\Omega }$ equal those computed using metrics on ${\mathrm{\Omega }}^{\mathrm{\prime }}$.

(vi) Nondegeneracy and scaling. $A(\rho )=0$ if and only if $\rho =0$ almost everywhere. If $0<A(\rho )<+\mathrm{\infty }$, then for every real scalar $c>0$ the quotient ${\mathrm{\ell }}_{c\rho }(\mathrm{\Gamma }{)}^2\mathrm{/}A(c\rho )$ equals ${\mathrm{\ell }}_{\rho }(\mathrm{\Gamma }{)}^2\mathrm{/}A(\rho )$. This includes ${\mathrm{\ell }}_{\rho }(\mathrm{\Gamma })=0$ and ${\mathrm{\ell }}_{\rho }(\mathrm{\Gamma })=+\mathrm{\infty }$; the denominator is always finite and positive.

• Christopher J. Bishop, Quasiconformal Mappings, Ch. 2 §2, printed pp. 51–53. Bishop defines a quadrilateral as a Jordan domain with two disjoint closed boundary arcs marked, assigns its modulus by a conformal rectangle, and defines geometric quasiconformality by the two-sided modulus bound for every quadrilateral.
• Mikhail Lyubich, Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I, Ch. 2 §12.5, printed p. 188. QC2 states that moduli of quadrilaterals and annuli are $K$-quasi-invariant for an orientation-preserving homeomorphism.
• Lars Ahlfors and Arne Beurling, Conformal Invariants and Function-Theoretic Null-Sets, §§4–5, printed pp. 114–120, for the extremal-length convention and its conformal invariance.

Let $\mathrm{\Omega },{\mathrm{\Omega }}^{\mathrm{\prime }}\subseteq \mathbb{C}$ be complex domains and $f:\mathrm{\Omega }\to {\mathrm{\Omega }}^{\mathrm{\prime }}$ a homeomorphism (A complex domain is a nonempty connected open subset of $\mathbb{C}$, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

Orientation. For $a\in \mathrm{\Omega }$, $f$ induces an isomorphism $f_{\ast }:H_2(\mathrm{\Omega },\mathrm{\Omega }\setminus \{a\};\mathbb{Z})⟶H_2({\mathrm{\Omega }}^{\mathrm{\prime }},{\mathrm{\Omega }}^{\mathrm{\prime }}\setminus \{f(a)\};\mathbb{Z})\mathrm{.}$ Excision and the local-homology calculation identify both groups with $\mathbb{Z}$ (Excision for singular homology, Local homology detects manifold dimension, interior, and boundary). The generators are those determined by the standard orientation of $\mathbb{C}\cong {\mathbb{R}}^2$, with positively oriented basis $(1,i)$ (Orientation of a finite-dimensional real vector space, R-orientation of a topological manifold); the restriction isomorphisms from coordinate balls to points are supplied by Coordinate-ball classes identify local homology stalks. Write ${\epsilon }_f(a)\in \{+1,-1\}$ for the multiplier of $f_{\ast }$. Naturality of relative homology (Functoriality of relative homology) makes these maps compatible with coordinate-ball restrictions. In the local-system charts of R-orientation of a topological manifold, their multiplier is locally constant. Since $\mathrm{\Omega }$ is connected, ${\epsilon }_f$ is constant. The map $f$ is orientation-preserving when this sign is $+1$, and orientation-reversing when it is $-1$.

A quadrilateral. A quadrilateral in $\mathrm{\Omega }$ is a set $Q=\phi (\bar{\mathrm{\Pi }})$, where $\mathrm{\Pi }=(0,w)\times (0,h)$ with $w,h>0$ and $\phi :\bar{\mathrm{\Pi }}\to \mathbb{C}$ is continuous and injective. Its two marked sides are either $\phi (\{0\}\times [0,h])$ and $\phi (\{w\}\times [0,h])$, or the other pair of opposite sides. For one such choice, let $\mathrm{\Gamma }(Q)$ be the family of continuous paths in $Q$ whose endpoints lie on different marked sides and whose interior lies in $\mathrm{int}Q$. This is a curve family in $\mathrm{\Omega }$; its modulus $\mu (\mathrm{\Gamma }(Q))$ is defined in Extremal length and the curve-family modulus of a path family. The image $f(Q)$ is again a quadrilateral with marked sides carried by $f$, and $f\mathrm{\Gamma }(Q)=\mathrm{\Gamma }(f(Q))$.

Geometric definition. Let $K\ge 1$. The homeomorphism $f$ is $K$-geometrically quasiconformal when it is orientation-preserving and, for every quadrilateral $Q$ with $\bar{Q}\subseteq \mathrm{\Omega }$ and for each choice of its marked sides, $\frac{1}{K}\mu (\mathrm{\Gamma }(Q))\le \mu (\mathrm{\Gamma }(f(Q)))\le K\mu (\mathrm{\Gamma }(Q)),$ using the conventions of Extremal length and the curve-family modulus of a path family for zero and infinite values. It is geometrically quasiconformal if it is $K$-geometrically quasiconformal for some finite $K$; the least such $K$ is $K_f\ge 1$, its maximal dilatation.

The modulus condition uses the extremal-length construction and its Countable-Choice hypothesis; the orientation sign itself uses no choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). The inverse of a $K$-geometrically quasiconformal map is also $K$-geometrically quasiconformal: its local-homology map is the inverse isomorphism, and the two modulus inequalities rearrange to the same bounds for $f^{-1}$. Thus $K_{f^{-1}}=K_f$. The local orientation clause and the modulus inequality are distinct parts of this definition.

• Christopher J. Bishop, Quasiconformal Mappings, Ch. 1 §1, printed pp. 2–4. Lemma 1.1 proves conformal invariance by transforming lengths and areas and then applying the inverse map; Lemma 1.2 proves overflow monotonicity; Lemma 1.3 and Corollary 1.4 give the disjoint-support modulus addition rule; Lemma 1.5 gives the series inequality.
• Mikhail Lyubich, Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I, Ch. 1 §§6.1–6.2, printed pp. 119–121. The definition identifies extremal width with the infimum of area over metrics whose length on every curve is at least one. The text then proves conformal invariance, the series law by normalizing two metrics to have equal length, area and extremal length, and the parallel law by restriction to disjoint supporting sets.
• Lars Ahlfors and Arne Beurling, Conformal Invariants and Function-Theoretic Null-Sets, §4, printed p. 115. Lemmas 1–3 state overflow monotonicity, the series inequality, and the parallel harmonic-sum law. The arguments below retain the extended-value and finite-positive-area conventions of this library.

Assume Countable Choice, and let $\lambda$ and $\mu$ be the extremal length and curve-family modulus of Extremal length and the curve-family modulus of a path family. Let $\mathrm{\Omega },{\mathrm{\Omega }}^{\mathrm{\prime }}\subseteq \mathbb{C}$ be complex domains. A path family in a domain consists of paths whose full traces lie in that domain.

(a) Conformal invariance. If $f:\mathrm{\Omega }\to {\mathrm{\Omega }}^{\mathrm{\prime }}$ is a biholomorphism (Biholomorphic maps between complex domains) and $\mathrm{\Gamma }$ is a path family in $\mathrm{\Omega }$, then $f\mathrm{\Gamma }:=\{f\circ \gamma :\gamma \in \mathrm{\Gamma }\}$ satisfies $\lambda (f\mathrm{\Gamma })=\lambda (\mathrm{\Gamma })$ and $\mu (f\mathrm{\Gamma })=\mu (\mathrm{\Gamma })$. Thus these parameters are invariants of conformal equivalence (Conformal equivalence and the automorphism group of a domain).

(b) Monotonicity and overflow. If $\mathrm{\Gamma }\subseteq {\mathrm{\Gamma }}^{\mathrm{\prime }}$, then $\lambda ({\mathrm{\Gamma }}^{\mathrm{\prime }})\le \lambda (\mathrm{\Gamma })$. If every $\gamma \in {\mathrm{\Gamma }}_0$ contains a subpath belonging to ${\mathrm{\Gamma }}_1$, then $\lambda ({\mathrm{\Gamma }}_0)\ge \lambda ({\mathrm{\Gamma }}_1)$ and $\mu ({\mathrm{\Gamma }}_0)\le \mu ({\mathrm{\Gamma }}_1)$.

(c) Series law. Let ${\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_1,\mathrm{\Gamma }$ be path families in $\mathrm{\Omega }$. Suppose there are disjoint Borel sets $E_0,E_1\subseteq \mathrm{\Omega }$ such that every path of ${\mathrm{\Gamma }}_j$ has trace in $E_j$, and every path in $\mathrm{\Gamma }$ has restrictions to two disjoint closed parameter intervals that belong respectively to ${\mathrm{\Gamma }}_0$ and ${\mathrm{\Gamma }}_1$. Then $\lambda (\mathrm{\Gamma })\ge \lambda ({\mathrm{\Gamma }}_0)+\lambda ({\mathrm{\Gamma }}_1)\mathrm{.}$

(d) Parallel law (Grötzsch). If ${\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_1$ are path families in $\mathrm{\Omega }$ and their traces lie respectively in disjoint Borel sets $E_0,E_1\subseteq \mathrm{\Omega }$, then $\mu ({\mathrm{\Gamma }}_0\cup {\mathrm{\Gamma }}_1)=\mu ({\mathrm{\Gamma }}_0)+\mu ({\mathrm{\Gamma }}_1)\mathrm{.}$ Equivalently, $\lambda ({\mathrm{\Gamma }}_0\cup {\mathrm{\Gamma }}_1)$ is the harmonic sum of $\lambda ({\mathrm{\Gamma }}_0)$ and $\lambda ({\mathrm{\Gamma }}_1)$, where the harmonic sum of $x,y\in [0,+\mathrm{\infty }]$ means $1\mathrm{/}(1\mathrm{/}x+1\mathrm{/}y)$ with the reciprocal conventions of the definition.

All assertions use the extended-real conventions of Extremal length and the curve-family modulus of a path family, in particular $\lambda (\mathrm{\varnothing })=+\mathrm{\infty }$, $\mu (\mathrm{\varnothing })=0$, and addition of $+\mathrm{\infty }$ to a nonnegative value gives $+\mathrm{\infty }$.