Calabi conjecture

Conjectures that have been proved | Complex manifolds | Theorems in differential geometry

In the mathematical field of differential geometry, the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by Eugenio Calabi . It was proved by Shing-Tung Yau , who received the Fields Medal and Oswald Veblen Prize in part for his proof. His work, principally an analysis of an elliptic partial differential equation known as the complex Monge–Ampère equation, was an influential early result in the field of geometric analysis. More precisely, Calabi's conjecture asserts the resolution of the prescribed Ricci curvature problem within the setting of Kähler metrics on closed complex manifolds. According to Chern–Weil theory, the Ricci form of any such metric is a closed differential 2-form which represents the first Chern class. Calabi conjectured that for any such differential form R, there is exactly one Kähler metric in each Kähler class whose Ricci form is R. (Some compact complex manifolds admit no Kähler classes, in which case the conjecture is vacuous.) In the special case that the first Chern class vanishes, this implies that each Kähler class contains exactly one Ricci-flat metric. These are often called Calabi–Yau manifolds. However, the term is often used in slightly different ways by various authors — for example, some uses may refer to the complex manifold while others might refer to a complex manifold together with a particular Ricci-flat Kähler metric. This special case can equivalently be regarded as the complete existence and uniqueness theory for Kähler–Einstein metrics of zero scalar curvature on compact complex manifolds. The case of nonzero scalar curvature does not follow as a special case of Calabi's conjecture, since the 'right-hand side' of the Kähler–Einstein problem depends on the 'unknown' metric, thereby placing the Kähler–Einstein problem outside the domain of prescribing Ricci curvature. However, Yau's analysis of the complex Monge–Ampère equation in resolving the Calabi conjecture was sufficiently general so as to also resolve the existence of Kähler–Einstein metrics of negative scalar curvature. The third and final case of positive scalar curvature was resolved in the 2010s, in part by making use of the Calabi conjecture. (Wikipedia).

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