Calabi conjecture

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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 by 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.

More formally, the Calabi conjecture states:

If M is a compact Kähler manifold with Kähler metric g\; and Kähler form \omega\;, and R is any (1,1)-form representing the manifold's first Chern class, then there exists a unique Kähler metric \tilde{g} on M with Kähler form \tilde{\omega} such that \omega\; and \tilde{\omega} represent the same class in cohomology H2(M,R) and the Ricci form of \tilde{\omega} is R.

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

Kähler–Einstein metrics[edit]

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.

It was disproved for positive first Chern classes by Yau, who observed that the complex projective plane blown up at 2 points has no Kähler–Einstein metric and so is a counterexample. Also even when Kähler–Einstein metric exists it need not be unique. There has been a lot of further work on the positive first Chern class case. A necessary condition for the existence of a Kähler–Einstein metric is that the Lie algebra of holomorphic vector fields is reductive. Yau conjectured that when the first Chern class is positive, a Kähler variety has a Kähler–Einstein metric if and only if it is stable in the sense of geometric invariant theory.

The case of complex surfaces has been settled by Gang Tian. The complex surfaces with positive Chern class are either a product of two copies of a projective line (which obviously has a Kähler–Einstein metric) or a blowup of the projective plane in at most 8 points in "general position", in the sense that no 3 lie on a line and no 6 lie on a quadric. The projective plane has a Kähler–Einstein metric, and the projective plane blown up in 1 or 2 points does not, as the Lie algebra of holomorphic vector fields is not reductive. Tian showed that the projective plane blown up in 3, 4, 5, 6, 7, or 8 points in general position has a Kähler–Einstein metric.

Outline of the proof of the Calabi conjecture[edit]

Calabi transformed the Calabi conjecture into a non–linear partial differential equation of complex Monge–Ampere 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.

Transformation of the Calabi conjecture to a differential equation[edit]

Suppose that M is a complex compact manifold with a Kahler form ω. Any other Kahler form in the same class is of the form

\omega+dd'\phi

for some smooth function φ on M, unique up to addition of a constant. The Calabi conjecture is therefore equivalent to the following problem:

Let F=ef be a positive smooth function on M with average value 1. Then there is a smooth real function φ with
(\omega+dd'\phi)^m = e^f\omega^m
and φ is unique up to addition of a constant.

This is an equation of complex Monge–Ampere type for a single function φ. It is a particularly hard partial differential equation to solve, as it is non-linear in the terms of highest order. It is trivial to solve it when f=0, as φ=0 is a solution. The idea of the continuity method is to show that it can be solved for all f by showing that the set of f for which it can be solved is both open and closed. Since the set of f for which it can be solved is non-empty, and the set of all f is connected, this shows that it can be solved for all f.

The map from smooth functions to smooth functions taking φ to F defined by

F=(\omega+dd'\phi)^m/\omega^m

is neither injective nor surjective. It is not injective because adding a constant to φ does not change F, and it is not surjective because F must be positive and have average value 1. So we consider the map restricted to functions φ that are normalized to have average value 0, and ask if this map is an isomorphism onto the set of positive F=ef with average value 1. Calabi and Yau proved that it is indeed an isomorphism. This is done in several steps, described below.

Uniqueness of the solution[edit]

Proving that the solution is unique involves showing that if

(\omega+dd'\varphi_1)^m = (\omega+dd'\varphi_2)^m

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

|d(\varphi_1-\varphi_2)|^2

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

d(\varphi_1-\varphi_2) = 0

which in turn forces φ1 and φ2 to differ by a constant.

The set of F is open[edit]

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.

The set of F is closed[edit]

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.

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