A cornerstone of Kähler geometry is Yau’s celebrated proof of the Calabi conjecture, which implies

We now give a very sketchy account of (the modern view on) Yau’s proof of Theorem 4.1. Fix a background Kähler metric $\omega$ in the given class, and the task is to find a Kähler potential $\phi$ solving the complex Monge-Ampère equation for any given density function $e^{f}$ satisfying the cohomological constraint $\int_{X}e^{f}\omega^{n}=\int_{X}\omega^{n}$,

$$\omega_{\phi}^{n}=(\omega+dd^{c}\phi)^{n}=e^{f}\omega^{n},\quad\int_{X}\phi\omega^{n}=0,$$

(9)

In particular for $e^{f}\omega^{n}=\text{const}\cdot\Omega\wedge\overline{\Omega}$ one obtains the Calabi-Yau metric. The uniqueness follows from a simple integration by parts argument.

The existence proof follows the continuity method, namely to deform through the space of Kähler metrics as $f$ deforms from zero to the desired function. Viewing (9) as defining a map from the potential to the density function, we need to show the map is submersive and proper. The submersion property is a consequence of standard Hodge theory on the Laplacian. To show properness one needs a priori estimates on $\phi$ depending only on $X$ and bounds on $f$, and for that matter we are free to impose any order of regularity on $f$.

The first step is to bound $\left\lVert\phi\right\rVert_{C^{0}}$ (known as ‘$C^{0}$-estimate’), using a technique called Moser iteration. We rewrite the equation as

$$(1-e^{f})\omega^{n}=-dd^{c}\phi\wedge(\omega^{n-1}+\omega^{n-1}\wedge\omega_{\phi}+\ldots+\omega_{\phi}^{n-1}).$$

For any $p>1$, multiply the equation by $\phi|\phi|^{p-2}$ and integrate by parts,

$$\int_{X}|\nabla|\phi|^{p/2}|_{\omega}^{2}\omega^{n}\leq\frac{C(n,\left\lVert f\right\rVert_{L^{\infty}})p^{2}}{p-1}\int_{X}|\phi|^{p-1}\omega^{n}.$$

The key is that a weaker norm on RHS controls a stronger norm on LHS. This is reverse to the direction of the Poincaré inequality

$$\int_{X}u^{2}\omega^{n}\leq C\int_{X}|du|_{\omega}^{2}\omega^{n},\quad\int_{X}u=0,$$

and the Sobolev inequality

$$\int_{X}|u|^{\frac{2n}{n-1}}\omega^{n}\leq C(\int_{X}|du|^{2}_{\omega}\omega^{n}+\int_{X}|u|^{2}\omega^{n}).$$

The combined force is that $\left\lVert\phi\right\rVert_{L^{2}_{1}}\leq C$, and one can inductively bound $\left\lVert\phi\right\rVert_{L^{p}}$ for large $p$, which turns out to be uniform in $p$. Taking the limit $p\to\infty$, this gives a bound $\left\lVert\phi\right\rVert_{L^{\infty}}\leq C(X,\left\lVert f\right\rVert_{L^{\infty}})$.

The second step is to bound $\left\lVert dd^{c}\phi\right\rVert_{C^{0}}$ (known as $C^{1,\bar{1}}$-estimate), i.e. to prove a uniform equivalence between $\omega$ and $\omega_{\phi}$. The starting point is a differential inequality by local calculations (in the analyst’s convention of Laplacian)

$$\Delta_{\omega_{\phi}}\log\Tr_{\omega}\omega_{\phi}\geq-C\Tr_{\omega_{\phi}}\omega-C.$$

Subtracting a large enough multiply of the identity

$$\Delta_{\omega_{\phi}}\phi=n-\Tr_{\omega_{\phi}}\omega,$$

we get a differential inequality

$$\Delta_{\omega_{\phi}}(\log\Tr_{\omega}\omega_{\phi}-C\phi)\geq\Tr_{\omega_{\phi}}\omega-C^{\prime}.$$

Using the a priori bound on $\phi$, an application of maximum principle then shows $\Tr_{\omega}\omega_{\phi}\leq C$.

The third step is to get higher order estimates. Standard elliptic theory implies it is sufficient to have a $C^{2,\alpha}$ bound on $\phi$. Evans-Krylov theory bridged the gap between $C^{1,\bar{1}}$-bound and $C^{2,\alpha}$-bound (cf. [66] Chapter 2, Section 4 for details). This argument is of local nature, and crucially uses that $\log\det(\partial_{i}\partial_{\bar{j}}\phi)$ is a concave function of the matrix $(\partial_{i}\partial_{\bar{j}}\phi)$, and the main tool is a Harnack inequality. The geometric insight is that higher order regularity is a manifestation of the local Euclidean nature of Kähler manifolds.

Yau’s proof strategy is highly influential and permeates the vast majority of works on CY metrics. The brief sketch above, however, highlights two reasons why it is difficult to adapt to the setting of SYZ conjecture:

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  The CY metrics undergoing large complex structure limit are highly degenerate, to the extent that the Sobolev constant becomes too big, so that the Moser iteration technique does not give useful $C^{0}$-estimate on the potential.

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  The $C^{1,\bar{1}}$ bound has global nature, so in order to obtain useful estimates on the Calabi-Yau metric only in the generic region, the design of the maximum principle must hold globally. This is hard on highly degenerate manifolds, where Riemannian curvature cannot be globally uniformly bounded.

For these reasons, our approach to the SYZ conjecture has significant departure from Yau’s proof. The potential estimates use complex pluripotential theory instead, and the metric estimate in the generic region uses a deep theorem of Savin in elliptic PDE theory, bypassing Yau’s estimates.