Calabi conjecture

Suppose that ${\displaystyle M}$ is a complex compact manifold with a Kähler form ${\displaystyle \omega }$ Any other Kähler form in the same class is of the form

${\displaystyle \omega +dd'\varphi }$

for some smooth function ${\displaystyle \varphi }$ on ${\displaystyle M}$, unique up to addition of a constant. The Calabi conjecture is therefore equivalent to the following problem:

Let ${\displaystyle F=e^{f}}$ be a positive smooth function on ${\displaystyle M}$ with average value 1. Then there is a smooth real function ${\displaystyle \varphi }$; with

${\displaystyle (\omega +dd'\varphi )^{m}=e^{f}\omega ^{m}}$

and ${\displaystyle \varphi }$; is unique up to addition of a constant.

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

The map from smooth functions to smooth functions taking ${\displaystyle \varphi }$ to ${\displaystyle F}$ defined by

${\displaystyle F=(\omega +dd'\varphi )^{m}/\omega ^{m}}$

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

Proving that the solution is unique involves showing that if

${\displaystyle (\omega +dd'\varphi _{1})^{m}=(\omega +dd'\varphi _{2})^{m}}$

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

${\displaystyle |d(\varphi _{1}-\varphi _{2})|^{2}}$

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

${\displaystyle d(\varphi _{1}-\varphi _{2})=0}$

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

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 φ_i and their higher derivatives in terms of the higher derivatives of log(f_i). 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 φ_i 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.