Conformal invariants and quadratic differentials

A general principle in complex function theory is that functionals on classes of conformal maps are associated to quadratic differentials. Various theorems and techniques illustrate this principle. For example, the variational theory of M. Schiffer shows that extremal mappings map onto the image domain minus trajectories of a quadratic differential. In this talk, we associate functionals on bounded univalent functions to quadratic differentials with singularities. This functional is conformally invariant, monotonic, and bounded. Examples of particular inequalities (new and old) are given, arising from various choices of the quadratic differential.