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The universal Teichm\"uller space is an infinitely dimensional generalizationof the classical Teichm\"uller space of Riemann surfaces. It carries a naturalHilbert structure, on which one can define a natural Riemannian metric, theWeil-Petersson metric. In this paper we investigate the Weil-PeterssonRiemannian curvature operator $\tilde{Q}$ of the universal Teichm\"uller spacewith the Hilbert structure, and prove the following:  (i) $\tilde{Q}$ is non-positive definite.  (ii) $\tilde{Q}$ is a bounded operator.  (iii) $\tilde{Q}$ is not compact; the set of the spectra of $\tilde{Q}$ isnot discrete.  As an application, we show that neither the Quaternionic hyperbolic space northe Cayley plane can be totally geodesically immersed in the universalTeichm\"uller space endowed with the Weil-Petersson metric.

The Weil-Petersson curvature operator on the universal Teichm\"uller space