Theorem 12.3.12220E0C47510

Proof. Concavity was proved in proposition 12.2.2. Suppose first that $n\geq 2$ and that equality holds. Take a minimizing Gauduchon metric for the middle background $(1-s)\theta _0+s\theta _1$. Its quotient at the middle background is the same convex combination of its quotients at $\theta _0$ and $\theta _1$, and each of these is at least the corresponding infimum. Equality therefore forces this metric to minimize both endpoint quotients.

By theorem 12.2.3, the solution forms $\Omega _ i=\Omega _{\theta _ i}$, with constants $c_ i=c_{\theta _ i}(\mu )$ for $i=0,1$, are then both conformal to this metric. Their volume forms are $c_0\mu $ and $c_1\mu $, so their pointwise proportionality factor is the constant $r=(c_1/c_0)^{1/n}$. Hence $\Omega _1=r\Omega _0$, which gives the stated smooth $\mathrm{dd}^{\mathrm{c}}$ relation. Conversely, if that relation holds, invariance under smooth $\mathrm{dd}^{\mathrm{c}}$ changes and homogeneity give equality.

When $n=1$, the function is $\theta \mapsto \int _ X\theta $. Take $r=\int _ X\theta _1/\int _ X\theta _0$. The difference $\theta _1-r\theta _0$ has integral zero, so the Green operator of the compact curve writes it as $\mathrm{dd}^{\mathrm{c}}f$ with $f$ smooth. Thus the same equivalence holds. $\square$