Conformal modulus

Remarks

The modulus is the reciprocal of the extremal length: $M(\Gamma) = 1/\lambda(\Gamma)$, with $1/0 = \infty$ and $1/\infty = 0$. It is invariant under conformal maps of $\Omega$. Example: for an annulus $\set{r < \abs{z} < R}$ and the family $\Gamma$ of curves in it that join the two boundary circles, $\rho(z) = \frac{1}{\abs{z}\log(R/r)}$ is admissible, and $M(\Gamma) = \frac{2\pi}{\log(R/r)}$. The check of the lower bound integrates the inequality $\int_r^R \rho(te^{i\theta})\, dt \ge 1$ over $\theta$ and applies the Cauchy–Schwarz inequality. Non-example: $\rho = \frac{1}{2\abs{z}\log(R/r)}$ is not admissible, since along a radial segment its integral is $\frac{1}{2}$. A quadrilateral is a Jordan domain $Q$ with four marked boundary points, which cut the boundary into four sides. Its modulus is the modulus of the family of curves in $Q$ that join one pair of opposite sides. Map $Q$ conformally onto a rectangle $(0, a) \times (0, b)$, with the marked points at the corners and the chosen sides vertical; then the modulus is $b/a$. Conventions for ring domains differ: the modulus of $\set{r < \abs{z} < R}$ is often defined as $\frac{1}{2\pi}\log(R/r)$, which is the extremal length of the family above, or as $\log(R/r)$.