Milin's coefficients, complex geometry of Teichm\"{u}ller spaces and variational calculus for univalent functions

TL;DR

This paper explores the geometry of Teichmüller spaces using Milin's inequalities, showing all invariant metrics coincide with the Teichmüller metric and deriving new distortion results for univalent functions.

Contribution

It establishes the equivalence of invariant metrics with the Teichmüller metric in certain spaces and applies this to variational theory for univalent functions with quasiconformal extensions.

Findings

All non-expanding invariant metrics coincide with the Teichmüller metric.

Deep distortion results for classes of univalent functions.

Identification of new phenomena in geometric function theory.

Abstract

We investigate the invariant metrics and complex geodesics in the universal Teichm\"{u}ller space and Teichm\"{u}ller space of the punctured disk using Milin's coefficient inequalities. This technique allows us to establish that all non-expanding invariant metrics in either of these spaces coincide with its intrinsic Teichm\"{u}ller metric. Other applications concern the variational theory for univalent functions with quasiconformal extension. It turns out that geometric features caused by the equality of metrics and connection with complex geodesics provide deep distortion results for various classes of such functions and create new phenomena which do not appear in the classical geometric function theory.