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When no smooth Gauduchon metric attains the threshold, a minimizer can still be found among weak limits of normalized Gauduchon forms. The normalization bounds the coefficient measures and hence gives compactness. Let $\mathscr G_\omega $ be the set of positive $(n-1,n-1)$-currents $T$ on $X$ satisfying
\[ \mathrm{dd}^{\mathrm{c}}T=0,\qquad \int _ X\omega \wedge T=1. \]
We call them normalized Gauduchon currents. Let $\mathscr C_\omega $ be the weak closure of the set of smooth strictly positive members of $\mathscr G_\omega $. By the root calculation preceding the proof of theorem 11.6.1, these smooth members are exactly the normalized powers $\gamma ^{n-1}/\int _ X\omega \wedge \gamma ^{n-1}$ of Gauduchon metrics $\gamma $.
Proposition 11.7.1.155386134D9E The sets $\mathscr G_\omega $ and $\mathscr C_\omega $ are nonempty, compact and convex. For every smooth real $(1,1)$-form $\theta $,
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11.7.2
\begin{equation} \kappa _\omega (\theta ) =\min _{T\in \mathscr C_\omega }\int _ X\theta \wedge T. \end{equation}
Moreover, a current $T\in \mathscr G_\omega $ belongs to $\mathscr C_\omega $ if and only if
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11.7.3
\begin{equation} \int _ X\theta \wedge T\geq 0 \quad \text{for every smooth real }\theta \text{ with }\mathrm{PSH}(X,\theta )\ne \varnothing . \end{equation}
Proof.
Gauduchon's theorem gives a smooth strictly positive member of $\mathscr G_\omega $ after normalization. The fixed trace mass bounds the total variation of all coefficient measures in finitely many local frames, so weak compactness of finite measures gives compactness of the family of positive currents with this trace. Positivity, the trace normalization and the equation $\mathrm{dd}^{\mathrm{c}}T=0$ pass to weak limits. On this bounded set, testing against smooth forms or against forms with continuous coefficients gives the same topology, by uniform approximation, and this topology is metrizable since the space of continuous test forms is separable. Thus the compactness can also be used sequentially. The defining conditions of $\mathscr G_\omega $ are convex. Its smooth strictly positive members form a convex set, so their closure $\mathscr C_\omega $ is compact and convex as well.
Pairing with a fixed smooth $\theta $ is weakly continuous. Its minimum on the closure of the smooth members is therefore their infimum, which proves (11.7.2). The Lamari criterion shows that (11.7.3) holds for every smooth strictly positive member, and hence for every member of their weak closure.
Conversely, suppose that $T\in \mathscr G_\omega $ does not belong to $\mathscr C_\omega $, and separate this point from the compact convex set $\mathscr C_\omega $ in the space of currents with the weak topology. A continuous linear functional for that topology is evaluation against a smooth form. After changing its sign if necessary, we obtain a smooth real $(1,1)$-form $\beta $ such that
\[ \int _ X\beta \wedge T <\min _{R\in \mathscr C_\omega }\int _ X\beta \wedge R =\kappa _\omega (\beta ). \]
By corollary 11.6.2, the smooth form $\beta -\kappa _\omega (\beta )\omega $ admits a positive current. Its pairing with $T$ is negative, because $\int _ X\omega \wedge T=1$. This contradicts (11.7.3) and proves the converse.
$\square$
Corollary 11.7.4.A97DA8CE2358 The function $\kappa _\omega $ is concave, positively homogeneous and invariant under the addition of $\mathrm{dd}^{\mathrm{c}}f$ for $f\in C^\infty (X,\mathbb R)$. Let $\theta $ and $\eta $ be smooth real $(1,1)$-forms and let $\varepsilon \geq 0$. If $-\varepsilon \omega \leq \theta -\eta \leq \varepsilon \omega $, then
\[ |\kappa _\omega (\theta )-\kappa _\omega (\eta )|\leq \varepsilon . \]
If smooth real $(1,1)$-forms $\theta _ j$ converge uniformly to a smooth form $\theta $ and $T_ j\in \mathscr C_\omega $ minimize their pairings, every weak cluster point of $(T_ j)$ minimizes the pairing with $\theta $.
Proof.
By (11.7.2), $\kappa _\omega $ is an infimum of linear functionals, which proves concavity and positive homogeneity. Invariance under the addition of $\mathrm{dd}^{\mathrm{c}}f$ follows from $\mathrm{dd}^{\mathrm{c}}T=0$. The normalization of the trace gives
\[ \left|\int _ X(\theta -\eta )\wedge T\right|\leq \varepsilon \quad (T\in \mathscr C_\omega ), \]
and taking infima proves the estimate. For the last assertion, pass to a subsequence $T_ j\rightharpoonup T\in \mathscr C_\omega $. The estimate and the uniform convergence of the backgrounds give
\[ \int _ X\theta \wedge T =\lim _ j\int _ X\theta _ j\wedge T_ j =\lim _ j\kappa _\omega (\theta _ j)=\kappa _\omega (\theta ). \]
Thus $T$ is a minimizer.
$\square$
In general, the closure $\mathscr C_\omega $ cannot be replaced by all of $\mathscr G_\omega $: the exceptional-curve example in example 10.18.3 separates the two sets by a smooth closed form. Testing against all of $\mathscr G_\omega $ instead detects smooth positivity and nefness, as shown in theorem 12.34.1.
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