## Journal of Differential Geometry

### The completion of the manifold of Riemannian metrics

Brian Clarke

#### Abstract

We give a description of the completion of the manifold of all smooth Riemannian metrics on a fixed smooth, closed, finitedimensional, orientable manifold with respect to a natural metric called the $L^2$ 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.

#### Article information

Source
J. Differential Geom., Volume 93, Number 2 (2013), 203-268.

Dates
First available in Project Euclid: 25 February 2013

https://projecteuclid.org/euclid.jdg/1361800866

Digital Object Identifier
doi:10.4310/jdg/1361800866

Mathematical Reviews number (MathSciNet)
MR3024306

Zentralblatt MATH identifier
1183.53003

#### Citation

Clarke, Brian. The completion of the manifold of Riemannian metrics. J. Differential Geom. 93 (2013), no. 2, 203--268. doi:10.4310/jdg/1361800866. https://projecteuclid.org/euclid.jdg/1361800866