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In mathematics, the Calabi conjecture is a conjecture about the existence of good metrics on complex manifolds, made by Calabi in about 1954. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, Eugenio Calabi (born 1923 is an Italian-American Mathematician and professor emeritus at the University of Pennsylvania, specializing in Differential The conjecture was proven by S. T. Yau in 1976. Shing-Tung Yau ( born April 4, 1949) is a Chinese American Mathematician working in Differential geometry, and involved

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 Mathematics, a Kähler manifold is a Manifold with unitary structure (a ''U''(''n''-structure) satisfying an Integrability condition In Mathematics, in particular in Algebraic topology and differential geometry, the Chern classes are a particular type of Characteristic 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. In Mathematics, in particular in Algebraic topology and differential geometry, the Chern classes are a particular type of Characteristic class In Differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined In mathematics Calabi&ndashYau manifolds are compact Kähler manifolds whose Canonical bundle is trivial

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

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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 T. Aubin and S. T. Yau in 1976. Thierry Aubin is a French mathematician at Centre de Mathématiques de Jussieu who works on Riemannian geometry and non-linear Partial differential equations Shing-Tung Yau ( born April 4, 1949) is a Chinese American Mathematician working in Differential geometry, and involved 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. Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane 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. In Mathematics Geometric invariant theory (or GIT) is a method for constructing quotients by group actions in Algebraic geometry, used to construct moduli

The case of complex surfaces has been settled by Gang Tian. Tian Gang ( 1958 &ndash  is a Chinese Mathematician and an Academician of the Chinese Academy of Sciences. 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

The conjecture was proved by transforming the problem into a non-linear partial differential equation of complex Monge-Ampere type, and then solving 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 the solution is proving certain "a priori" estimates for the derivatives of solutions; this was done by Yau, and most of the rest of the solution is due to Calabi.

Transformation of the Calabi conjecture to a differential equation

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

ω + dd'φ

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
(ω + dd'φ)m = efωm
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 = (ω + dd'φ)m / ωm

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 absolute value 1, 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

Proving that the solution is unique involves showing that if

(ω + dd'φ1)m = (ω + dd'φ2)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(φ1 − φ2) | 2

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

d(φ1 − φ2) = 0

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

The set of F is open

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 the branch of Mathematics called Multivariable calculus, the implicit function theorem is a tool which allows relations to be converted to functions In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis

The set of F is closed

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 bounds for the functions φi and their higher derivatives in terms of the higher derivatives of log(fi) (these are sometimes called "apriori bounds"). 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.

References

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