extremal length

How do you measure the 'shape' of a region in a way that does not change when you bend it conformally? Ordinary length and area change under stretching, but conformal maps preserve angles, so there ought to be a quantity that survives them. Extremal length is exactly such a conformal invariant: a way of assigning a number to a family of curves (say, all curves crossing a rectangle the short way) that is unchanged by any conformal map. It is the workhorse of the modern, metric-geometry approach to conformal mapping.

The idea, due to Ahlfors and Beurling, is a clever optimization. Given a family of curves in a region, you consider all ways of putting a conformal metric rho |dz| on the region (a non-negative weight rho), measure each curve's rho-length and the region's rho-area, and then take the supremum over metrics of (shortest rho-length of the family)^2 divided by (total rho-area). That supremum is the extremal length of the family. The metric that achieves it is the 'extremal metric', and it concentrates weight exactly where the curves are forced to be long. For a rectangle, the family of curves joining the two short sides has extremal length equal to the ratio (width / height) — recovering the familiar conformal modulus of the rectangle.

Because it is conformally invariant, extremal length turns geometric questions into computable comparisons: it gives clean proofs of distortion and growth estimates, bounds on harmonic measure, and the modulus of ring domains and quadrilaterals. It is also the natural language for quasiconformal maps, which distort extremal length by only a bounded factor. A caution: extremal length is a property of a CURVE FAMILY, not of a single curve or a single point, and the defining supremum is genuinely an extremal (optimization) problem — the name is literal. Computing it exactly is usually possible only in symmetric situations; in general one estimates it from above and below.