The mapping class group action on Teichmüller spaces of hyperkähler manifolds

The central result in the moduli theory of compact hyperkähler manifolds is Verbitsky's global Torelli theorem. It describes the Teichmüller space parametrizing complex structures on such manifolds as a non-Hausdorff covering of a certain homogeneous space called the period domain. The Teichmüller space carries a natural action of the mapping class group, i.e. the group of diffeomorphisms modulo diffeomorphisms isotopic to the identity. One can use Ratner's theory to describe the closures of the orbits of this action. In my talk I will try to give a general introduction into this circle of ideas and then explain how one can apply them. I will present some new results from my joint work with Sibony and Verbitsky concerning rigid currents on hyperkähler manifolds.