We show that a polarized affine variety with an isolated singularity admits a Ricci flat Kähler cone metric if and only if it is K–stable. This generalizes the Chen–Donaldson–Sun solution of the Yau–Tian–Donaldson conjecture to Kähler cones, or equivalently, Sasakian manifolds. As an application we show that the five-sphere admits infinitely many families of Sasaki–Einstein metrics.
"Sasaki–Einstein metrics and K–stability." Geom. Topol. 23 (3) 1339 - 1413, 2019. https://doi.org/10.2140/gt.2019.23.1339