Sasaki-Einstein metrics and K-stability

We show that a polarized affine variety admits a Ricci flat K"ahler cone metric, if and only if it is K-stable. This generalizes Chen-Donaldson-Sun’s solution of the Yau-Tian-Donaldson conjecture to K"ahler cones, or equivalently, Sasakian manifolds. As an application we show that the five-sphere admits infinitely many families of Sasaki-Einstein metrics.