We prove the existence of a Kähler–Einstein metric on a K–stable Fano manifold using the recent compactness result on Kähler–Ricci flows. The key ingredient is an algebrogeometric description of the asymptotic behavior of Kähler–Ricci flow on Fano manifolds. This is in turn based on a general finite-dimensional discussion, which is interesting on its own and could potentially apply to other problems. As one application, we relate the asymptotics of the Calabi flow on a polarized Kähler manifold to K–stability, assuming bounds on geometry.
"Kähler–Ricci flow, Kähler–Einstein metric, and K–stability." Geom. Topol. 22 (6) 3145 - 3173, 2018. https://doi.org/10.2140/gt.2018.22.3145