The Mabuchi Completion of the Space of Kähler Potentials

Abstract

Suppose $(X,\omega )$ is a compact Kähler manifold. Following Mabuchi, the space of smooth Kähler potentials $\mathcal{H}$ can be endowed with a Riemannian structure, which induces an infinite dimensional path length metric space $(\mathcal{H},d)$. We prove that the metric completion of $(\mathcal{H},d)$ can be identified with $({\mathcal{E}}^2(X,\omega ),\tilde{d})$, and this latter space is a complete non-positively curved geodesic metric space. In obtaining this result, we will rely on envelope techniques which allow for a treatment in a very general context. Profiting from this, we will characterize the pairs of potentials in $\text{PSH}(X,\omega )$ that can be connected by weak geodesics and we will also give a characterization of $\mathcal{E}(X,\omega )$ in this context.

arXiv commentv3 New title and new introduction. No other changes