Extremal length of the rectangle and of the round annulus

Put $P(s,\theta )=s(\cos \theta +i\sin \theta )$ on $(r,R)\times (0,2\pi )$. Its Jacobian determinant is $s>0$, so the inverse-function theorem in [F3] gives local $C^1$ inverses. The principal polar-form theorem gives a unique angle in $(0,2\pi )$ for every point off the positive real ray; thus $P$ is bijective onto the cut annulus and its local inverses combine to a global $C^1$ inverse. The omitted ray is null by [F3]. The nonnegative Borel change-of-variables theorem [F4] therefore gives the displayed polar area identity for every nonnegative Borel integrand, including extended-valued ones.

Let $\rho$ be any Borel density on $\mathrm{\Pi }$ with $0<A(\rho )<\mathrm{\infty }$, and put $L={\mathrm{\ell }}_{\rho }({\mathrm{\Gamma }}_{\mathrm{\Pi }}^{\mathrm{v}})$. For each $y\in (0,h)$ the horizontal segment from $(0,y)$ to $(w,y)$ belongs to ${\mathrm{\Gamma }}_{\mathrm{\Pi }}^{\mathrm{v}}$, so $L\le {\int }_0^w\rho (x,y)dx$. For almost every $y$, Tonelli and finite area make $\rho (\cdot ,y)$ square-integrable; Cauchy–Schwarz then gives $L^2\le w{\int }_0^w\rho (x,y{)}^{2}dx$. In particular $L<\mathrm{\infty }$. Integrating over $y$ yields $h{L}^2\le wA(\rho )$, so every quotient is at most $w\mathrm{/}h$.

Let $\rho$ be any Borel density on $A(r,R)$ with $0<A(\rho )<\mathrm{\infty }$, and put $L={\mathrm{\ell }}_{\rho }({\mathrm{\Gamma }}_{r,R})$. For each $\theta \in (0,2\pi )$ the radial segment $s\mapsto s{e}^{i\theta }$ belongs to ${\mathrm{\Gamma }}_{r,R}$, so $L\le {\int }_r^R\rho (s{e}^{i\theta })ds$. For almost every $\theta$, the square integral is finite; weighted Cauchy–Schwarz gives ${\displaystyle L^2\le \left({\int }_r^R\frac{ds}{s}\right){\int }_r^R\rho (s{e}^{i\theta }{)}^{2}sds=\log (R\mathrm{/}r){\int }_r^R\rho (s{e}^{i\theta }{)}^{2}sds\mathrm{.}}$ Integrating in $\theta$ and applying [F2, F4] yields $2\pi L^2\le \log (R\mathrm{/}r)A(\rho )$; in particular $L<\mathrm{\infty }$. Hence every quotient for ${\mathrm{\Gamma }}_{r,R}$ is at most $(2\pi {)}^{-1}\log (R\mathrm{/}r)$.

For a rectifiable path $\gamma$ joining the two annulus boundary circles, its compact trace lies in $\{\mathrm{\mid }z\mathrm{\mid }\ge r\}\subset {\mathbb{C}}^{\times }$. Cover the trace by discs avoiding $0$ and subdivide its parameter interval so each subpath lies in one such disc, using compactness and the Lebesgue number lemma (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset, Every open cover of a compact metric space has a Lebesgue number: a $\delta >0$ such that every nonempty subset of diameter less than $\delta$ lies inside a single member of the cover). On each disc take a holomorphic logarithm $G_j$ of $z$ by [F6]; differentiating $e^{G_j(z)}=z$ gives $G_j^{\mathrm{\prime }}(z)=1\mathrm{/}z$. Since $\mathrm{\mid }{e}^{G_j(z)}\mathrm{\mid }=e^{\mathrm{Re}{G}_j(z)}=\mathrm{\mid }z\mathrm{\mid }$, [F6, F7] identify $\mathrm{Re}{G}_j(z)$ with $\log \mathrm{\mid }z\mathrm{\mid }$. The fundamental theorem for complex line integrals [F5] evaluates each subpath integral, and additivity [F5] gives ${\displaystyle \mathrm{Re}{\int }_{\gamma }\frac{dz}{z}=\log \mathrm{\mid }\gamma (b)\mathrm{\mid }-\log \mathrm{\mid }\gamma (a)\mathrm{\mid }=\pm \log (R\mathrm{/}r)\mathrm{.}}$ The sign depends on the initial endpoint. The fundamental inequality in [F5] therefore yields ${\displaystyle {\mathrm{\ell }}_{1\mathrm{/}\mathrm{\mid }z\mathrm{\mid }}(\gamma )={\int }_{\gamma }\frac{\mathrm{\mid }dz\mathrm{\mid }}{\mathrm{\mid }z\mathrm{\mid }}\ge \mid {\int }_{\gamma }\frac{dz}{z}\mid \ge \log (R\mathrm{/}r)\mathrm{.}}$ For nonrectifiable paths the left side is $+\mathrm{\infty }$ by definition.

Put $S=R\mathrm{/}r$ and $D(z)=z\mathrm{/}r$. Both $D:A(r,R)\to A(1,S)$ and its inverse $w\mapsto rw$ have constant complex difference quotients, so they are holomorphic and $D$ is biholomorphic (Biholomorphic maps between complex domains, Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions). In the componentwise Riemann–Stieltjes definition, replacing $\gamma$ by $D\circ \gamma$ multiplies its coordinate integrators by $1\mathrm{/}r$, while the pulled-back integrand $1\mathrm{/}(D\circ \gamma )$ is multiplied by $r$; therefore ${\int }_{D\circ \gamma }dw\mathrm{/}w={\int }_{\gamma }dz\mathrm{/}z$. Hence $D$ maps ${\mathrm{\Theta }}_{r,R}$ bijectively onto ${\mathrm{\Theta }}_{1,S}$. Conformal invariance Conformal invariance, monotonicity, and the series and parallel laws for extremal length gives equality of their extremal lengths and moduli. It remains to calculate the normalized family in $A(1,S)$.

For the normalized winding-one family ${\mathrm{\Theta }}_{1,S}$, every $\gamma$ satisfies ${\int }_{\gamma }dz\mathrm{/}z=2\pi i$ by [F8]. The continuous density $\rho (z)=1\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$ therefore has ${\mathrm{\ell }}_{\rho }(\gamma )\ge 2\pi$ by [F1, F5]. Its area is ${\displaystyle A(\rho )={\int }_0^{2\pi }{\int }_1^S\frac{1}{s^2}sdsd\theta =2\pi \log S}$ by [F4, F7]. It is finite and positive, so $\lambda ({\mathrm{\Theta }}_{1,S})\ge 2\pi \mathrm{/}\log S$.

The constant density $\rho =1\mathrm{/}w$ on $\mathrm{\Pi }$ gives every path in ${\mathrm{\Gamma }}_{\mathrm{\Pi }}^{\mathrm{v}}$ length at least $1$, because its endpoint displacement is $w$ and Euclidean path length is at least that distance. Its area is $h\mathrm{/}w$, so its quotient is at least $w\mathrm{/}h$. Together with step 1.2 this gives $\lambda ({\mathrm{\Gamma }}_{\mathrm{\Pi }}^{\mathrm{v}})=w\mathrm{/}h$ and $\mu ({\mathrm{\Gamma }}_{\mathrm{\Pi }}^{\mathrm{v}})=h\mathrm{/}w$. Interchanging the two coordinates gives the horizontal formulas.

The Borel density $\rho (z)=1\mathrm{/}(2\pi \mathrm{\mid }z\mathrm{\mid })$ on $A(r,R)$ has ${\mathrm{\ell }}_{\rho }({\mathrm{\Gamma }}_{r,R})\ge (2\pi {)}^{-1}\log (R\mathrm{/}r)$ by step 1.4. Its area, by step 1.1 and [F7], is ${\displaystyle A(\rho )={\int }_0^{2\pi }{\int }_r^R\frac{1}{4{\pi }^2{s}^2}sdsd\theta =\frac{1}{2\pi }\log (R\mathrm{/}r),}$ which is finite and positive. Its quotient is therefore at least $(2\pi {)}^{-1}\log (R\mathrm{/}r)$. With step 1.3 this proves $\lambda ({\mathrm{\Gamma }}_{r,R})=(2\pi {)}^{-1}\log (R\mathrm{/}r)$ and $\mu ({\mathrm{\Gamma }}_{r,R})=2\pi \mathrm{/}\log (R\mathrm{/}r)$.

For any finite-positive-area Borel density $\rho$ on $A(1,S)$ put $L={\mathrm{\ell }}_{\rho }({\mathrm{\Theta }}_{1,S})$. For each $s\in (1,S)$ the circle ${\gamma }_s(\theta )=s{e}^{i\theta }$, $0\le \theta \le 2\pi$, has winding number $1$ by [F8], so $L\le {\mathrm{\ell }}_{\rho }({\gamma }_s)$. Its speed is $s$, hence its arc-length function is $s\theta$; the Stieltjes interval formula, finite-measure uniqueness, and increasing simple approximation give ${\displaystyle {\mathrm{\ell }}_{\rho }({\gamma }_s)=s{\int }_0^{2\pi }\rho (s{e}^{i\theta })d\theta \mathrm{.}}$ For almost every $s$, the finite area and [F4] make the circle density square-integrable, and Cauchy–Schwarz gives $L^2\le 2\pi s^2{\int }_0^{2\pi }\rho (s{e}^{i\theta }{)}^{2}d\theta$. Rearranging this as ${\displaystyle \frac{L^2}{2\pi s}\le s{\int }_0^{2\pi }\rho (s{e}^{i\theta }{)}^{2}d\theta }$ and integrating over $s\in (1,S)$ gives ${\displaystyle \frac{\log S}{2\pi }{L}^2\le {\int }_1^S{\int }_0^{2\pi }\rho (s{e}^{i\theta }{)}^{2}sd\theta ds=A(\rho )}$ by [F4]. Therefore every quotient is at most $2\pi \mathrm{/}\log S$, which with step 1.6 proves $\lambda ({\mathrm{\Theta }}_{1,S})=2\pi \mathrm{/}\log S$ and $\mu ({\mathrm{\Theta }}_{1,S})=(2\pi {)}^{-1}\log S$. Step 1.5 and $\log S=\log (R\mathrm{/}r)$ transfer these two values to ${\mathrm{\Theta }}_{r,R}$.

The positive rectangle segments, radial annulus segments, and once-traversed circles used above witness that all assigned families are nonempty. The formulas are finite and strictly positive because $w,h>0$ and $\log (R\mathrm{/}r)>0$ by [F7]. Taking reciprocals under the conventions of [F9] gives the four displayed modulus values, while steps 2.1, 2.2 and 2.3 give all four extremal lengths.