PomeBE24BBA15991Corollary 17.16.14.BE24BBA15991 Fix $u\in \mathcal H_\omega $ and a nonzero constant $a$. The path $u_ t=u+at$, $0\leq t\leq 1$, is a Perron segment and solves the homogeneous equation (17.16.10). It is stationary for the $L^2$ action if and only if
For $n\geq 2$, this means that $\omega _ u$ is a Gauduchon metric.
Proof. The path is admissible and equals the chord between its endpoints, so by the chord bound no admissible competitor can be larger; it is therefore the Perron segment. Its acceleration and the spatial derivatives of its velocity vanish, which proves (17.16.10). In (17.16.13), the only remaining term is $\frac n2a^2\mathrm{dd}^{\mathrm{c}}(\omega _ u^{n-1})$, which vanishes exactly under the stated condition. $\square$
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