# Doctoral thesis 3/2009

## The Completion of the Manifold of Riemannian Metrics with Respect to its L^{2} Metric

### Brian Clarke

**Contact the author:** Please use for correspondence this email.**Submission date: **07. Apr. 2009**Pages: 135****MSC-Numbers: **58D17, 58B20**Download full preprint:** PDF (1086 kB)**Abstract:**

This is the author's Ph.D. thesis, submitted to the University of Leipzig. It deals with the Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold.

The main body of the thesis is a description of the completion of the manifold of metrics with respect to the metric. The primary motivation for studying this problem comes from Teichmüller theory, where similar considerations lead to a completion of the well-known Weil-Petersson metric. We give an application of the main theorem to the completions of Teichmüller space with respect to a class of metrics that generalize the Weil-Petersson metric.

We also prove that the metric induces a metric space structure on the manifold of metrics. As the metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Frechet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finite-dimensional manifold or a strong Riemannian metric on a Hilbert manifold.