Let $X$ be a compact Kähler manifold with $c_1(X) = 0$ and let $[\omega_0]$ be a Kähler class. We seek a Kähler metric $\omega = \omega_0 + i\partial\bar{\partial}\varphi$ with $\text{Ric}(\omega) = 0$.

Step 1: Setup of the Monge-Ampère equation

The Ricci curvature of a Kähler metric satisfies: $\text{Ric}(\omega) = -i\partial\bar{\partial}\log(\omega^n)$

For $\omega = \omega_0 + i\partial\bar{\partial}\varphi$ to be Ricci-flat, we need: $(\omega_0 + i\partial\bar{\partial}\varphi)^n = e^F \omega_0^n$

where $F$ is determined by $c_1(X) = 0$. Since $c_1(X) = 0$, we have $\text{Ric}(\omega_0) = i\partial\bar{\partial}f$ for some function $f$, so $F = -f$. The equation becomes: $\det\left(g_{i\bar{j}} + \frac{\partial^2\varphi}{\partial z^i\partial\bar{z}^j}\right) = e^{-f} \det(g_{i\bar{j}})$

This is the complex Monge-Ampère equation with $\varphi$ as the unknown.

Step 2: Establishing a priori estimates

The core difficulty is obtaining a priori estimates for $\varphi$. We need bounds on:

• $C^0$ norm: $\|\varphi\|_{C^0} \leq C_0$
• Gradient: $|\nabla\varphi| \leq C_1$
• Higher derivatives: $\|\varphi\|_{C^k} \leq C_k$

The $C^0$ estimate follows from the maximum principle and the constraint $\int_X e^{-f+\varphi}\omega_0^n = \int_X \omega_0^n$.

For the gradient estimate, we use Yau's method: consider the quantity: $Q = \log\text{tr}_{\omega_0}(\omega) - A\varphi$

where $A$ is a large constant. At a maximum point of $Q$, the Laplacian $\Delta_{\omega_0}Q$ can be computed using the Monge-Ampère equation, yielding: $\Delta_{\omega_0}Q \geq -C$

for a constant $C$ independent of $\varphi$. This bounds $\text{tr}_{\omega_0}(\omega)$ and hence $|\nabla\varphi|$.

Step 3: Higher regularity

Once gradient bounds are established, higher derivative estimates follow from elliptic regularity theory. The linearization of the Monge-Ampère operator is elliptic when $\omega > 0$, allowing us to apply the Calderón-Zygmund theory and Schauder estimates.

Specifically, we obtain $C^{2,\alpha}$ estimates, then bootstrap to $C^\infty$ using the elliptic equation and standard regularity results.

Step 4: Continuity method

With a priori estimates in hand, we use the continuity method. Define a family of equations: $(\omega_0 + i\partial\bar{\partial}\varphi_t)^n = e^{tf} \omega_0^n, \quad t \in [0,1]$

At $t=0$, the solution is $\varphi_0 = 0$. We show the set $S = \{t \in [0,1] : \varphi_t \text{ exists}\}$ is both open and closed in $[0,1]$.

• Openness: By the implicit function theorem for the Monge-Ampère operator
• Closedness: By the a priori estimates ensuring convergence of solutions

Therefore $S = [0,1]$, and $\varphi_1$ gives the desired Ricci-flat metric.

Step 5: Uniqueness

Uniqueness follows from the maximum principle. If $\omega_1$ and $\omega_2$ are two Ricci-flat metrics in the same Kähler class, then $\omega_1 - \omega_2 = i\partial\bar{\partial}u$ for some function $u$. The difference of the Monge-Ampère equations gives: $\Delta_{\omega_1}u = 0$

By the maximum principle on compact $X$, $u$ is constant, so $\omega_1 = \omega_2$.