We start by constructing a Hilbert manifold T of orientation preserving
diffeomorphisms of the circle (modulo the group of bi-holomorphic self-mappings
of the disc). This space, which could be thought of as a completion of the
universal Teichmueller space, is endowed with a right-invariant Kaehler metric.
Using results from the theory of quasiconformal mappings we construct an
embedding of T into the infinite dimensional Segal-Wilson Grassmannian. The
latter turns out to be a very natural ambient space for T. This allows us to
prove that T's sectional curvature is negative in the holomorphic directions
and by a reasoning along the lines of Cartan-Hadamard's theory that its
geodesics exist for all time. The geodesics of T lead to solutions of the
periodic Korteweg-de Vries (KdV) equation by means of V. Arnold's
generalization of Euler's equation. As an application, we obtain long-time
existence of solutions to the periodic KdV equation with initial data in a
certain closed subspace of the periodic Sobolev space of index 3/2.