Hilbert Structure on the universal Teichmuller space and its Weil-Petersson curvature operator

Zeno Huang (CUNY)

The universal Teichmuller space is an infinitely dimensional complex Banach manifold which contains all classical Teichmuller spaces. To study its Riemannian geometry, Takhtajan and Teo (2006) introduced a Hilbert structure and generalized many results in Weil-Petersson geometry from Teichmuller space to the universal one. In joint work with Y. Wu of Tsinghua university, we investigate the curvature operator of this metric on this Hilbert manifold and prove that it's bounded, nonpositive definite and noncompact.