Calabi conjecture

Theorem about the existence of certain Riemannian metrics on complex manifolds

In mathematics, the Calabi conjecture was a conjecture about the existence of certain "nice" Riemannian metrics on certain complex manifolds, made by Eugenio Calabi (1954, 1957) and proved by Shing-Tung Yau (1977, 1978). Yau received the Fields Medal in 1982 in part for this proof.

The Calabi conjecture states that a compact Kähler manifold has a unique Kähler metric in the same class whose Ricci form is any given 2-form representing the first Chern class. In particular if the first Chern class vanishes there is a unique Kähler metric in the same class with vanishing Ricci curvature; these are called Calabi–Yau manifolds.

The Calabi conjecture is closely related to the question of which Kähler manifolds have Kähler–Einstein metrics.

A conjecture closely related to the Calabi conjecture states that if a compact Kähler variety has a negative, zero, or positive first Chern class then it has a Kähler–Einstein metric in the same class as its Kähler metric, unique up to rescaling. This was proved for negative first Chern classes independently by Thierry Aubin and Shing-Tung Yau in 1976. When the Chern class is zero it was proved by Yau as an easy consequence of the Calabi conjecture. These results were never explicitly conjectured by Calabi, but would have followed from results that he announced in his 1954 talk at the International Congress of Mathematicians.[citation needed]

When the first Chern class is positive, the above conjecture is actually false as a consequence of a result of Yozo Matsushima, which shows that the complex automorphism group of a Kähler–Einstein manifold of positive scalar curvature is necessarily reductive. For example, the complex projective plane blown up at 2 points has no Kähler–Einstein metric and so is a counterexample. Another problem arising from complex automorphisms is that they can lead to a lack of uniqueness for the Kähler–Einstein metric, even when it exists. However, complex automorphisms are not the only difficulty that arises in the positive case. Indeed, it was conjectured by Yau et al that when the first Chern class is positive, a Kähler manifold admits a Kähler–Einstein metric if and only if it is K-stable. A proof of this conjecture was published by Xiuxiong Chen, Simon Donaldson and Song Sun in January 2015,[1][2][3] and Tian gave a proof electronically published on September 16, 2015.[4][5]

On the other hand, in the special case of complex dimension two, a compact complex surface with positive first Chern class does admit a Kähler–Einstein metric if and only if its automorphism group if reductive. This important result is often attributed to Gang Tian. Since Tian’s proof, there have been some simplifications and refinements of arguments involved; cf. the paper by Odaka, Spotti, and Sun cited below. The complex surfaces that admit such Kähler–Einstein metrics are therefore exactly the complex projective plane, the product of two copies of a projective line, and blowups of the projective plane in 3 to 8 points in general position.[citation needed]

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. This involves first solving an easier equation, and then showing that a solution to the easy equation can be continuously deformed to a solution of the hard equation. The hardest part of Yau's solution is proving certain a priori estimates for the derivatives of solutions.

then φ1 and φ2 differ by a constant (so must be the same if they are both normalized to have average value 0). Calabi proved this by showing that the average value of

is given by an expression that is at most 0. As it is obviously at least 0, it must be 0, so

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.

This is the hardest part of the proof, and was the part done by Yau. Suppose that F is in the closure of the image of possible functions φ. This means that there is a sequence of functions φ1, φ2, ... such that the corresponding functions F1, F2,... converge to F, and the problem is to show that some subsequence of the φs converges to a solution φ. 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). Finding these bounds requires a long sequence of hard estimates, each improving slightly on the previous estimate. 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. This subsequence converges to a function φ with image F, which shows that the set of possible images F is closed.