Weil-Petersson geometry of Teichmüller spaces and the Thurston classification
of surface diffeomorphisms
In this talk I will discuss the geometry of Teichmüller spaces with
respect to a Riemannian metric, called the Weil-Petersson metric.
Compared to the better-studied Teichmüller (Finsler) metric, the Weil-Petersson
metric suffers from the defect of being incomplete. It will be demonstrated
that by taking the completion of the space, one obtains a more comprehensive
picture of the Teichmüller space. In particular the isometric
action of the mapping class group extends to the completion, providing
a reformulation of Thurston's classification of surface diffeomorphisms.
Spring 2001 meeting of the PNGS