The Mabuchi Completion of the Space of Kähler Potentials

Abstract

Suppose $(X,ω)$ 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 $(\mathcal E^2(X,ω),\tilde d)$, 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 $\text{PSH}(X,ω)$ that can be connected by weak geodesics and we will also give a characterization of $\mathcal E(X,ω)$ in this context.