The Hilbert 3/2 Structure and Weil-Petersson Metric on the Space of the Diffeomorphisms of the Circle Modulo Conformal Maps

The space of diffeomorphisms of the circle modulo boundary values of conformal maps of the disk. The Sobolev 3/2 norm on the tangent space space at id of the space of diffeomorphisms of the circle modulo boundary values of conformal maps of the disk induces the only Kähler left invariant metric defined up to a constant on the space of diffeomorphisms of the circle modulo boundary values of conformal maps of the disk. We proved before that the completion of the holomorphic tangent space at the identity of the space of diffeomorphisms of the circle modulo boundary values of conformal maps of the disk with respect to the Sobolev 3/2 norm can be embedded as a closed Hilbert susbspace into the tangent space at a point of the Segal-Wilson Grassmannian. It was established in our previous paper that the natural metric on the Segal-Wilson Grassmannian The author was supported by Max-Plank Institute fur Mathematik, Bonn during the preparation of this paper.