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

TL;DR

This paper studies the Weil-Petersson curvature operator on the universal Teichmüller space, proving it is non-positive, bounded, and non-compact, with implications for embeddings of certain symmetric spaces.

Contribution

It establishes fundamental properties of the Weil-Petersson curvature operator on the universal Teichmüller space, a significant step in understanding its geometric structure.

Findings

The curvature operator is non-positive definite.

The operator is bounded but not compact.

The spectrum of the operator is not discrete.

Abstract

The universal Teichm\"uller space is an infinitely dimensional generalization of the classical Teichm\"uller 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 $\tilde{Q}$ of the universal Teichm\"uller space with 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 $\tilde{Q}$ 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 Teichm\"uller space endowed with the Weil-Petersson metric.