PomeCorollary 12.10.14.6362B224AE69 Suppose $\omega $ is Kähler. For a nonzero nonnegative $f\in L^1(\mathrm{d}V)$, the functional $D_{f^{1/n}}$ has a smooth strictly positive maximizer in $\mathscr G_\omega $ if and only if $f$ agrees almost everywhere with a smooth strictly positive function.
Corollary 12.10.146362B224AE69
Proof. The Kähler formula in proposition 15.28.7 gives $b_\omega (f)>0$. If a smooth strictly positive maximizer exists, the proof of corollary 12.10.12 gives
\[ \omega +\mathrm{dd}^{\mathrm{c}}u=a\gamma , \qquad a\geq 0, \]
with $a,u$ smooth and $a\not\equiv 0$. Since $\mathrm{d}\omega =0$, the first-order equation in lemma 5.22.4 becomes $\mathrm{d}a=-aC$ for a smooth one-form $C$. Along any smooth path $\sigma $, this is the linear scalar equation
\[ (a\circ \sigma )'=-(a\circ \sigma )C(\dot\sigma ). \]
whose solution is the initial value multiplied by a positive exponential. Since $X$ is connected and $a\not\equiv 0$, it follows that $a>0$ everywhere, and $f=b_\omega (f)^{-1}a^ n\gamma ^ n/\mathrm{d}V$ is smooth and strictly positive. Conversely, if $f$ is smooth and strictly positive, the smooth equation and theorem 12.2.3 provide the maximizing Gauduchon form. $\square$
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