Metric geometry of normal K\"ahler spaces, energy properness, and existence of canonical metrics

Let $(X,\omega )$ be a compact normal K"ahler space, with Hodge metric $\omega$. In this paper, the last in a sequence of works studying the relationship between energy properness and canonical K"ahler metrics, we introduce a geodesic metric structure on ${\mathcal{H}}_{\omega }(X)$, the space of K"ahler potentials, whose completion is the finite energy space $\mathcal E^1_{\omega}(X)$. Using this metric structure and the results of Berman-Boucksom-Eyssidieux-Guedj-Zeriahi as ingredients in the existence/properness principle of Rubinstein and the author, we show that existence of K"ahler-Einstein metrics on log Fano pairs is equivalent to properness of the K-energy in a suitable sense. To our knowledge, this result represents the first characterization of general log Fano pairs admitting K"ahler-Einstein metrics. We also discuss the analogous result for K"ahler-Ricci solitons on Fano varieties.