The Green function of Teichm\"uller spaces with applications

Abstract

We describe briefly a new approach to some problems related to Teichm\"uller spaces, invariant metrics, and extremal quasiconformal maps. This approach is based on the properties of plurisubharmonic functions, especially of the plurisubharmonic Green function. The main theorem gives an explicit representation of the Green function for Teichm\"uller spaces by the Kobayashi-Teichm\"uller metric of these spaces. This leads to various applications. In particular, this gives a new characterization of extremal quasiconformal maps.