Hyperbolic metrics on universal Teichmüller space and extremal problems (Q690519)

The main result of the paper under review is a simpler proof of the fact that the Carathéodory and Kobayashi metrics coincide on universal Teichmüller space. This is a result originally proved by the author [Sib. Mat. Zh. 45, No. 4, 780--808 (2004); translation in Sib. Math. J. 45, No. 4, 646--668 (2004; Zbl 1132.30349)]. These metrics are the smallest and largest metrics respectively which reduce distances under holomorphic mappings. As is well-known, the Kobayashi and Teichmüller metrics are equal on every Teichmüller space, but the equality of Carathéodory and Kobayashi metrics is only known on universal Teichmüller space. The author uses the methods of this proof of the main result to solve general extremal problems for univalent functions with quasiconformal extensions.