In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The most important special case of these are the Calabi–Yau manifolds, which are Kähler and Ricci-flat. The most important problem for this area is the existence of Kähler–Einstein metrics for compact Kähler manifolds. This problem can be split up into three cases dependent on the sign of the first Chern class of the Kähler manifold: When the first Chern class is negative, there is always a Kähler–Einstein metric, as Thierry Aubin and Shing-Tung Yau proved independently. When the first Chern class is zero, there is always a Kähler–Einstein metric, as Yau proved in the Calabi conjecture. That leads to the name Calabi–Yau manifolds. He was awarded with the Fields Medal partly because of this work. The third case, the positive or Fano case, remained a well-known open problem for many years. In this case, there is a non-trivial obstruction to existence. In 2012, Xiuxiong Chen, Simon Donaldson, and Song Sun proved that in this case existence is equivalent to an algebro-geometric criterion called K-stability. Their proof appeared in a series of articles in the Journal of the American Mathematical Society. A proof was produced independently by Gang Tian at the same time. When first Chern class is not definite, or we have intermediate Kodaira dimension, then finding canonical metric remained as an open problem, which is called the algebrization conjecture via analytical minimal model program. Definition Einstein manifolds Suppose is a Riemannian manifold. In physics the Einstein field equations are a set of partial differential equations on the metric tensor which describe how the manifold should curve due to the existence of mass or energy, a quantity encapsulated by the stress–energy tensor . In a vacuum where there is no mass or energy, that is , the Einstein Field Equations simplify. Namely, the Ricci curvature of is a symmetric -tensor, as is the metric itself, and the equations reduce to where is the scalar curvature of . That is, the Ricci curvature becomes proportional to the metric. A Riemannian manifold satisfying the above equation is called an Einstein manifold. Every two-dimensional Riemannian manifold is Einstein. It can be proven using the Bianchi identities that, in any larger dimension, the scalar curvature of any