The Completion of the Manifold of Riemannian Metrics

TL;DR

This paper characterizes the completion of the space of smooth Riemannian metrics on a closed manifold using the $L^2$ metric, with applications to Teichmüller theory and generalizations of the Weil-Petersson metric.

Contribution

It provides a detailed description of the metric completion of the manifold of Riemannian metrics and applies this to Teichmüller space with generalized metrics.

Findings

Completion of the manifold of Riemannian metrics is characterized.

Application to Teichmüller space with generalized metrics.

Connections to Weil-Petersson metric completion.

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 $L^2$ metric. The primary motivation for studying this problem comes from Teichmueller 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 Teichmueller space with respect to a class of metrics that generalize the Weil-Petersson metric.