The Mabuchi completion of the space of Kähler potentials

Suppose $(X,\omega)$ is a compact K\"ahler manifold. Following Mabuchi, the space of smooth K\"ahler potentials ${\cal H}$ can be endowed with a Riemannian structure, which induces an infinite dimensional path length metric space $({\cal H},d)$. We prove that the metric completion of $({\cal H},d)$ can be identified with $({\cal E}^2(X,\omega),{\tilde d})$, 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 ${\rm PSH}(X,\omega)$ that can be connected by weak geodesics and we will also give a characterization of ${\cal E}(X,\omega)$ in this context.