12.3 The equality case of concavity and the first variation484879715C58

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. $\square$

Proof. Choose a smooth potential $u_*$ with $\theta +\mathrm{dd}^{\mathrm{c}}u_*>0$. Backgrounds $\theta _ t$ near $\theta $ then have the positive representatives $\theta _ t+\mathrm{dd}^{\mathrm{c}}u_*$. The implicit-function argument in the proof of proposition 11.40.4 applies to this family and shows that the solution and its constant depend smoothly on the parameters. Adding $u_*$ to these solution potentials and fixing the mean gives the normalized solutions for $\theta _ t$, so $\lambda _\mu (\theta _ t)=c_{\theta _ t}(\mu )^{1/n}$ is smooth.

To compute the derivative, write $\lambda =\lambda _\mu (\theta )$ and $\Omega =\Omega _\theta $, and write the minimizing metric as $\gamma _\theta =\mathrm{e}^ g\Omega $ with $g$ smooth and real, as theorem 12.2.3 permits. Along $\theta _ t=\theta +t\chi $ for small real $t$, put $\Omega _ t=\theta _ t+\mathrm{dd}^{\mathrm{c}}u_ t$ and $\lambda _ t=\lambda _\mu (\theta _ t)$, and let dots denote derivatives at $t=0$. Differentiating $\Omega _ t^ n=\lambda _ t^ n\mu $ gives

We multiply by $\mathrm{e}^{(n-1)g}$, so that $\Omega ^{n-1}$ becomes $\gamma _\theta ^{n-1}$, and integrate. The term with $\mathrm{dd}^{\mathrm{c}}\dot u$ vanishes by the Gauduchon condition, and $\int _ X\chi \wedge \gamma _\theta ^{n-1}=\lambda ^{n-1}\dot\lambda \int _ X\mathrm{e}^{(n-1)g}\mu $. On the other hand, $\gamma _\theta ^ n=\mathrm{e}^{ng}\lambda ^ n\mu $, so that

Dividing the two identities proves the derivative formula. In dimension one the formula follows directly from the definition. $\square$