The homogeneous equation on the product involves the spatial form $\eta _ t$ only pointwise. In the $L^2$ action, integration by parts also differentiates the powers of $\eta _ t$, which produces terms absent from that equation. Fix a Hermitian form $\omega $, write $\omega _ u=\omega +\mathrm{dd}^{\mathrm{c}}u$, and recall from section 11.1 the space $\mathcal H_\omega $ of smooth real functions $u$ with $\omega _ u>0$. On $\mathcal H_\omega $ consider the Riemannian metric
\[ G_ u(v,w)=\int _ Xvw\, \omega _ u^ n, \qquad v,w\in C^\infty (X,\mathbb R). \]
For a smooth path $u_ t\in \mathcal H_\omega $, $0\leq t\leq 1$, its action is
\[ \mathscr S(u)=\frac12\int _0^1\int _ X\dot u_ t^{\, 2}\, \omega _{u_ t}^ n\, \mathrm{d}t. \]
Stationarity under smooth variations fixing both endpoints is the formal geodesic equation of this metric.
Theorem 17.16.12.7A7AE56CA513 Let $u_ t\in \mathcal H_\omega $, $0\leq t\leq 1$, be a smooth path, and put $\eta _ t=\omega _{u_ t}$ and $T_ t=\eta _ t^{n-1}$. The path is stationary for $\mathscr S$ if and only if
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17.16.13
\begin{equation} \begin{split} \ddot u_ t\, \eta _ t^ n={}& n\, \mathrm{d}\dot u_ t\wedge \mathrm d^{\mathrm c}\dot u_ t\wedge T_ t\\ & +n\dot u_ t\bigl (\mathrm{d}\dot u_ t\wedge \mathrm d^{\mathrm c}T_ t -\mathrm d^{\mathrm c}\dot u_ t\wedge \mathrm{d}T_ t\bigr ) +\frac n2\dot u_ t^{\, 2}\mathrm{dd}^{\mathrm{c}}T_ t. \end{split} \end{equation}
All differentials on the right act only in the $X$ variable.
Proof.
Vary the path as $u_ t+s v_ t$, where $v_ t$ is smooth in $(x,t)$ and $v_0=v_1=0$, and put $f=\dot u_ t$. Differentiating the action at $s=0$, using $\partial _ s(\omega _{u_ t+sv_ t}^ n)|_{s=0}=n\mathrm{dd}^{\mathrm{c}}v_ t\wedge T_ t$, gives
\[ \delta \mathscr S =\int _0^1\left(\int _ X f\dot v_ t\, \eta _ t^ n +\frac n2\int _ X f^2\mathrm{dd}^{\mathrm{c}}v_ t\wedge T_ t\right)\mathrm{d}t. \]
We integrate by parts in $t$ in the first term, using $\partial _ t(\eta _ t^ n)=n\mathrm{dd}^{\mathrm{c}}f\wedge T_ t$, and on $X$ in the second, using lemma 11.1.2 with the whole form $f^2T_ t$. Thus
\[ \delta \mathscr S =\int _0^1\int _ Xv_ t\left(-\ddot u_ t\eta _ t^ n -nf\mathrm{dd}^{\mathrm{c}}f\wedge T_ t+\frac n2\mathrm{dd}^{\mathrm{c}}(f^2T_ t)\right)\mathrm{d}t. \]
The product rule gives
\[ \begin{split} \mathrm{dd}^{\mathrm{c}}(f^2T_ t)={}& 2f\mathrm{dd}^{\mathrm{c}}f\wedge T_ t +2\mathrm{d}f\wedge \mathrm d^{\mathrm c}f\wedge T_ t\\ & +2f\bigl (\mathrm{d}f\wedge \mathrm d^{\mathrm c}T_ t-\mathrm d^{\mathrm c}f\wedge \mathrm{d}T_ t\bigr ) +f^2\mathrm{dd}^{\mathrm{c}}T_ t. \end{split} \]
After the term $nf\mathrm{dd}^{\mathrm{c}}f\wedge T_ t$ cancels, the integrand is $v_ t$ times the difference between the right-hand and left-hand sides of (17.16.13). Since $\eta _ t>0$ on the compact set $X\times [0,1]$, every smooth $v_ t$ gives an admissible variation for small $|s|$, and the fundamental lemma of the calculus of variations gives both implications.
$\square$
For a Kähler background, $\mathrm{d}T_ t=\mathrm d^{\mathrm c}T_ t=0$, and the equation reduces to (17.16.10). The Gauduchon condition $\mathrm{dd}^{\mathrm{c}}T_ t=0$ on $\eta _ t$ removes only the last term. Translations by constants isolate that term and give an exact test for stationarity.
Corollary 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
\[ \mathrm{dd}^{\mathrm{c}}(\omega _ u^{n-1})=0. \]
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.
$\square$
The three-dimensional torus in example 11.23.5 carries a Gauduchon form $\omega $ and a smooth potential $u\in \mathcal H_\omega $ for which $\omega _ u$ is not Gauduchon. Thus $u_ t=u+t$ is a smooth Perron segment that is not stationary for the $L^2$ action. On a surface, the Gauduchon condition persists under perturbation of the potential by proposition 11.23.4, but it does not cancel the terms in (17.16.13) involving first derivatives of $T_ t$. On the Hopf background, basic potentials give a complete metric whose geodesics are the full Perron segments. Its relation to the raw action reveals a further distinction between stationarity and minimization.
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