coordinates by the formula By assumption and are in the same cohomology class , so the -lemma from Hodge theory implies there exists a smooth function such that . Any other metric is related to by a Kähler potential such that . It then follows that if is the Ricci form with respect to , then Thus to make we need to find such that This will certainly be true if the same equation is proven after removing the derivatives , and in fact this is an equivalent equation by the -lemma up to changing by the addition of a constant function. In particular, after removing and exponentiating, the equation is transformed into This partial differential equation is similar to a real Monge–Ampere equation, and is known as a complex Monge–Ampere equation, and subsequently can be studied using tools from convex analysis. Its behaviour is highly sensitive to the sign of the topological constant . The solutions of this equation appear as critical points of the K-energy functional introduced by Toshiki Mabuchi on the space of Kähler potentials in the class . Existence The existence problem for Kähler–Einstein metrics can be split up into three distinct cases, dependent on the sign of the topological constant . Since the Kähler form is always a positive differential form, the sign of depends on whether the cohomology class is positive, negative, or zero. In algebraic geometry this is understood in terms of the canonical bundle of : if and only if the canonical bundle is an ample line bundle, and if and only if is ample. If is a trivial line bundle, then . When the Kähler manifold is compact, the problem of existence has been completely solved. The case c1(X)<0 When the Kähler manifold satisfies the topological assumption , the canonical bundle is ample and so must be negative. If the necessary topological assumption is satisfied, that is there exists a Kähler metric such that , then it was proven by Aubin and Yau that a Kähler–Einstein always exists. The existence of a Kähler metric satisfying the topological assumption is a consequence of Yau's proof of the Calabi conjecture. Theorem (Aubin, Yau): A compact Kähler manifold with always admits a Kähler–Einstein metric. The case c1(X)=0 When the canonical bundle is trivial, so that , the manifold is said to be Calabi–Yau. These manifolds are of special significance in physics, where they should