  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.
