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.
"The completion of the manifold of Riemannian metrics." J. Differential Geom. 93 (2) 203 - 268, February 2013. https://doi.org/10.4310/jdg/1361800866