Extremal length is a concept from the field of complex analysis and geometric topology, specifically concerning the study of Riemann surfaces and conformal mappings. It is used to measure the size of families of curves on a surface and has applications in various areas, including Teichmüller theory and the study of conformal structures. Mathematically, the extremal length of a family of curves is defined via a certain optimization problem.
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For a path family on a Riemann surface, take the supremum of over measurable conformal metrics of finite positive area. This is unchanged under conformal equivalence, and a quasiconformal map of maximal dilatation changes it by a factor between and .