Ricci flow and the metric completion of the space of Kähler metrics

We consider the space of Kähler metrics as a Riemannian submanifold of the space of Riemannian metrics, and study the associated submanifold geometry. In particular, we show that the intrinsic and extrinsic distance functions are equivalent. We also determine the metric completion of the space of Kähler metrics, making contact with recent generalizations of the Calabi-Yau Theorem due to Dinew and Guedj-Zeriahi. As an application, we obtain a new analytic stability criterion for the existence of a Kähler-Einstein metric on a Fano manifold in terms of the Ricci flow and the distance function. We also prove that the Kähler-Ricci flow converges as soon as it converges in the metric sense.