The Completion of the Manifold of Riemannian Metrics

Brian Clarke
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Submission date: 07. Apr. 2009
Pages: 45
paper submitted to: Inventiones mathematicae
MSC-Numbers: 58D17, 58B20
Download preprint: PDF (490 kB)

Abstract:
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 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.