In the complex case, the (Aubin-Mabuchi) energy functional is defined as follows. Fix a smooth semipositive reference metric $\phi_{0}$ and set

$$E(\phi):=\frac{1}{n+1}\sum_{j=0}^{n}\int_{{X^{\mathrm{an}}}}(\phi-\phi_{0})(dd^{c}\phi)^{j}\wedge(dd^{c}\phi_{0})^{n-j}.$$

(7.1)

for any smooth metric $\phi$. Here $(dd^{c}\phi)^{j}\wedge(dd^{c}\phi_{0})^{n-j}$ is a mixed Monge-Ampère measure. It is a positive measure if $\phi$ is semipositive.

In the non-Archimedean case, mixed Monge-Ampère measures can be defined using intersection theory when $\phi$ and $\phi_{0}$ are model metrics, and the energy of $\phi$ is then defined exactly as above.

For two smooth/model metrics $\phi$, $\psi$ we have

$$E(\phi)-E(\psi)=\frac{1}{n+1}\sum_{j=0}^{n}\int_{{X^{\mathrm{an}}}}(\phi-\psi)(dd^{c}\phi)^{j}\wedge(dd^{c}\psi)^{n-j}.$$

(7.2)

This is proved using integration by parts in the complex case and follows from basic intersection theory in the non-Archimedean case.

We can draw two main conclusions from (7.2). First, the derivative of the energy functional is the Monge-Ampère operator, in the sense that

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

(7.3)

for a smooth/model metric $\phi$ on ${L^{\mathrm{an}}}$ and a smooth/model function $f$ on ${X^{\mathrm{an}}}$.

Second, $E(\psi)\geq E(\phi)$ when $\psi\geq\phi$ are semipositive. It then makes sense to set

$$E(\phi):=\inf\{E(\psi)\mid\psi\geq\phi,\ \text{$\psi$ a semipositive smooth/model metric on ${L^{\mathrm{an}}}$}\}.$$

for any singular semipositive metric $\phi\in\operatorname{PSH}({L^{\mathrm{an}}})$. The resulting functional

$$E:\operatorname{PSH}({L^{\mathrm{an}}})\to[-\infty,\infty)$$

has many good properties: $E$ is concave, monotonous, and satisfies $E(\phi+c)=E(\phi)+c$ for $c\in{\mathbf{R}}$. Further, $E$ is usc and continuous along decreasing nets.

The energy functional singles out a class ${\mathcal{E}}^{1}({L^{\mathrm{an}}})$ of metrics with finite energy, $E(\phi)>-\infty$. This class has good properties. In particular, one can (with some effort) define mixed Monge-Ampère measures $(dd^{c}\phi)^{j}\wedge(dd^{c}\psi)^{n-j}$ for $\phi,\psi\in{\mathcal{E}}^{1}({L^{\mathrm{an}}})$, and (7.1) continues to hold.

Let us now go back to the variational approach to solving the Monge-Ampère equation. Fix a positive measure $\mu$ on ${X^{\mathrm{an}}}$ of mass $(L^{n})$. In the complex case we assume $\mu$ is absolutely continuous with respect to Lebesgue measure, with density in $L^{p}$ for some $p>1$. In the non-Archimedean case we assume that $\mu$ is supported on some dual complex. In both cases, one can show that the functional $\phi\to\int(\phi-\phi_{0})\,\mu$ is (finite and) continuous on $\operatorname{PSH}({L^{\mathrm{an}}})$, where $\phi_{0}$ is the same reference metric as in (7.1). Thus the functional $F_{\mu}\colon\operatorname{PSH}({L^{\mathrm{an}}})\to[-\infty,\infty)$ defined by

$$F_{\mu}(\phi):=E(\phi)-\int(\phi-\phi_{0})\mu$$

is upper semicontinuous. It follows from (7.2) that $F_{\mu}$ does not depend on the choice of reference metric $\phi_{0}$. We also have $F_{\mu}(\phi+c)=F_{\mu}(\phi)$ for $\phi\in\operatorname{PSH}({L^{\mathrm{an}}})$, $c\in{\mathbf{R}}$. Thus $F_{\mu}$ descends to an usc functional on the quotient space $\operatorname{PSH}({L^{\mathrm{an}}})/{\mathbf{R}}$. By Theorem 6.1, the latter space is compact, so we can find $\phi\in\operatorname{PSH}({L^{\mathrm{an}}})$ maximizing $F_{\mu}$. It is clear that $\phi\in{\mathcal{E}}^{1}({L^{\mathrm{an}}})$, so the mixed Monge-Ampère measures of $\phi$ and $\phi_{0}$ are well defined. However, equation (7.3) no longer makes sense, since there is no reason for the metric $\phi+tf$ to be semipositive for $t\neq 0$. Therefore, it is not clear that $\operatorname{MA}(\phi)=\mu$, as desired. In the next section, we explain how to get around this problem.