Geodesic Rays and Kähler-Ricci Trajectories on Fano Manifolds

Abstract: Suppose $(X,J,\omega)$ is a Fano manifold and $t \to r_t$ is a diverging K\"ahler-Ricci trajectory. We construct a bounded geodesic ray $t \to u_t$ weakly asymptotic to $t \to r_t$, along which Ding's $\mathcal F$-functional decreases, partially confirming a folklore conjecture. In absence of non-trivial holomorphic vector fields this proves the equivalence between geodesic stability of the $\mathcal F$-functional and existence of K\"ahler-Einstein metrics. We also explore applications of our construction to Tian's $\alpha$-invariant.