Unique extremality
Let \(R\) be a Riemann surface and \(T(R)\) be the Teichmüller space of \(R\), viewed as the Teichmüller equivalence classes of Beltrami coefficients. The following theorems are proved in this paper: A Beltrami coefficient \(\mu\) is uniquely extremal in its Teichmüller class if, and only if, it is uniquely extremal in its infinitesimal Teichmüller class. Let \(\varphi\) be a nonzero holomorphic quadratic differential on \(R\) and let \(f\) be a quasiconformal mapping of \(R\) with Beltrami coefficient \(\mu=k|\varphi|/\varphi\), \(0<k<1\). Then \(f\) is uniquely extremal if and only if there exists a sequence \(\varphi_n\) of holomorphic quadratic differentials of finite norm such that (1) \(\varphi_n(z)\) converges to \(\varphi(z)\) uniformly on compact subsets of \(R\), and (2) \(k\|\varphi_n\|-\text{Re}\int_R\varphi_n\mu\to 0\). As an application of these theorems, an example of a uniquely extremal Beltrami coefficient with non-constant modulus is given. This solves a well-known problem which was open for a long period.
scientific article; zbMATH DE number 1221961
scientific article