Proposition 12.11.2854AAD5CF1CA

Proof. Since $D_\mu (\gamma )>0$, a metric $\gamma $ attaining the value zero satisfies $\int _ X\theta \wedge \gamma ^{n-1}=0$. By lemma 11.6.3, including its dimension-one case, $\theta $ is then smoothly $\mathrm{dd}^{\mathrm{c}}$-exact. Conversely, if $\theta =\mathrm{dd}^{\mathrm{c}}f$, Stokes' theorem makes every quotient zero, so every admissible metric attains the infimum. $\square$