We shall present a unified approach to solving the complex and non-Archimedean Monge-Ampère equations in any dimension. The method goes back to Alexandrov’s work in convex geometry [Ale38]. It was adapted to the complex case in [BBGZ13] and to the non-Archimedean analogue in [BFJ15].

The general strategy is to construct an energy functional

$$E:\operatorname{PSH}^{0}({L^{\mathrm{an}}})\to{\mathbf{R}}$$

whose derivative is the Monge-Ampère operator, $E^{\prime}=\operatorname{MA}$, in the sense that

$$\frac{d}{dt}E(\phi+tf)|_{t=0}=\int_{{X^{\mathrm{an}}}}f\operatorname{MA}(\phi),$$

for every continuous semipositive metric $\phi\in\operatorname{PSH}^{0}({L^{\mathrm{an}}})$ and every smooth/model function $f$ on ${X^{\mathrm{an}}}$.

Grant the existence of this functional for the moment. Given a measure $\mu$ on ${X^{\mathrm{an}}}$, consider the functional $F_{\mu}:\operatorname{PSH}^{0}({L^{\mathrm{an}}})\to{\mathbf{R}}$ defined by

$$\ F_{\mu}(\phi)=E(\phi)-\int\phi\,\mu.$$

Suppose we can find $\phi\in\operatorname{PSH}^{0}({L^{\mathrm{an}}})$ that maximizes $F_{\mu}$. Since the derivative of $F_{\mu}$ is equal to $F_{\mu}^{\prime}=\operatorname{MA}-\mu$, we then have $0=F_{\mu}^{\prime}(\phi)=\operatorname{MA}(\phi)-\mu$ as required.

Now, there are at least three problems with this approach:

• (1)

  There is a priori no reason why a maximizer should exist in $\operatorname{PSH}^{0}({L^{\mathrm{an}}})$. We resolve this by introducing a larger space $\operatorname{PSH}({L^{\mathrm{an}}})$ with suitable compactness properties and find a maximizer there.

• (2)

  Granted the existence of a maximizer $\phi\in\operatorname{PSH}({L^{\mathrm{an}}})$, we are maximizing over a convex set rather than a vector space, so there is no reason why $F_{\mu}^{\prime}(\phi)=0$. Compare maximizing the function $f(x)=x^{2}$ on the real interval $[-1,1]$: the maximum is not at a critical point.

• (3)

  In the end we want to show that—after all—the maximizer is continuous, that is, $\phi\in C^{0}({L^{\mathrm{an}}})$.

We shall discuss how to address (1) and (2) in the next two sections. The continuity result in (3) requires a priori capacity estimates due to Kołodziej, and will not be discussed in these notes.