K\"ahler-Einstein metrics along the smooth continuity method

We show that if a Fano manifold $M$ is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then $M$ admits a K\"ahler-Einstein metric. This is a strengthening of the solution of the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-Donaldson-Sun, and can be used to obtain new examples of K\"ahler-Einstein manifolds. We also give analogous results for twisted K\"ahler-Einstein metrics and Kahler-Ricci solitons.