Unique extremality

Extremal mappings have been one of the main topics in the theory of quasiconformal mappings since its earliest days, when Grötzsch solved the extremal problem for two rectangles. Grötzsch showed that among all quasiconformal mappings from a rectangle R_1 onto another rectangle R_2, mapping the sides of R_1 onto the corresponding sides of R_2, there exists a unique mapping with minimal dilatation (see [Gr]). Later, Teichmüller [T] generalized Grotzsch's ideas and found many extremal quasiconformal mappings, including the uniquely extremal mapping f in the set of all quasiconformal mappings homotopic to a given sense-preserving homeomorphism between two compact Riemann surfaces R and S = f(R) of genus greater than 1. In a neighborhood of all but finitely many points on the compact Riemann surface R, the extremal map f could be expressed as a conformal map followed by an affine map followed by a conformal map. The Beltrami coefficient off is of the form k|φ|φ, where 0 ≤ k < 1 and φ is an integrable holomorphic quadratic differential.