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 (1954, 1957).
It was proved by Shing-Tung Yau (1977, 1978), 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.
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.
Calabi transformed the Calabi conjecture into a non-linear partial differential equation of complex Monge–Ampère type, and showed that this equation has at most one solution, thus establishing the uniqueness of the required Kähler metric.
Yau proved the Calabi conjecture by constructing a solution of this equation using the continuity method.
The Calabi conjecture is therefore equivalent to the following problem: This is an equation of complex Monge–Ampère type for a single function
It is a particularly hard partial differential equation to solve, as it is non-linear in the terms of highest order.
that are normalized to have average value 0, and ask if this map is an isomorphism onto the set of positive
Proving that the solution is unique involves showing that if then φ1 and φ2 differ by a constant (so must be the same if they are both normalized to have average value 0).
Proving that the set of possible F is open (in the set of smooth functions with average value 1) involves showing that if it is possible to solve the equation for some F, then it is possible to solve it for all sufficiently close F. Calabi proved this by using the implicit function theorem for Banach spaces: in order to apply this, the main step is to show that the linearization of the differential operator above is invertible.
In order to do this, Yau finds some a priori bounds for the functions φi and their higher derivatives in terms of the higher derivatives of log(fi).
The bounds Yau gets are enough to show that the functions φi all lie in a compact subset of a suitable Banach space of functions, so it is possible to find a convergent subsequence.