LAUSR

The universal Teichmüller space is an infinitely dimensional generalization of the classical Teichmüller space of Riemann surfaces. It carries a natural Hilbert structure, on which one can define a natural Riemannian metric, the Weil–Petersson metric. In this paper we investigate the Weil–Petersson Riemannian curvature operator $$\tilde{Q}$$Q~ of the universal Teichmüller space with the Hilbert structure, and prove the following:(i)$$\tilde{Q}$$Q~ is non-positive definite.(ii)$$\tilde{Q}$$Q~ is a bounded operator.(iii)$$\tilde{Q}$$Q~ is not compact; the set of the spectra of $$\tilde{Q}$$Q~ is not discrete. As an application, we show that neither the Quaternionic hyperbolic space nor the Cayley plane can be totally geodesically immersed in the universal Teichmüller space endowed with the Weil–Petersson metric.