I will describe Hilbert structure on the universal Teichmuller space where the Weil-Petersson metric is well defined (via Takhatajan-Teo). It has been shown that the WP metric is negatively curved and Einstein. Instead of curvature tensor, we consider its curvature operator and study some fundamental properties of this operator. In particular we show the operator is non-positive, bounded, and noncompact. This is based on joint work with Yunhui Wu.

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