On the Gerstenhaber-Rauch principle (Q1089124)

Suppose \(f^*(z)\) is a \(K^*\)-quasiconformal self-homeomorphism of the unit disk U, where \(K^*\) is the minimum possible value among all quasiconformal mappings of U with the same boundary values as \(f^*\). It is known [\textit{E. Reich}, Ann. Acad. Sci. Fenn. 10, 469-475 (1985; Zbl 0592.30027)] that \(K^*\) can be calculated by a variational principle involving mappings of U harmonic with respect to admissible weight functions. The authors examine the weight functions that correspond to the case when the extremum for the variational principle is attained, and characterize the corresponding mappings \(f^*\) as Teichmüller mappings corresponding to quadratic differentials with finite norm.