Tian's most renowned contribution to the Kähler-Einstein problem came in 1997. Yau had conjectured in the 1980s, based partly in analogy to the Donaldson-Uhlenbeck-Yau theorem, that existence of a Kähler-Einstein metric should correspond to stability of the underlying Kähler manifold in a certain sense of geometric invariant theory. It was generally understood, especially following work of Akito Futaki, that the existence of holomorphic vector fields should act as an obstruction to the existence of Kähler-Einstein metrics. Tian, in his 1997 article, gave concrete examples of Kähler manifolds which had no holomorphic vector fields and also no Kähler-Einstein metrics, showing that the ideal criterion lies deeper. Yau had proposed that, rather than holomorphic vector fields on the manifold itself, it should be relevant to study the deformations of projective embeddings of Kähler manifolds under holomorphic vector fields on projective space. This idea was modified by Tian, introducing the notion of K-stability and showing that any Kähler-Einstein manifold must be K-stable.