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MiS Preprint

The Completion of the Manifold of Riemannian Metrics

Brian Clarke


We give a description of the completion of the manifold of all smooth Riemannian metrics on a fixed smooth, closed, finite-dimensional, 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.

MSC Codes:
58D17, 58B20

Related publications

2013 Journal Open Access
Brian Clarke

The completion of the manifold of Riemannian metrics

In: Journal of differential geometry, 93 (2013) 2, pp. 203-268