Corollary 11.23.11.1C7A6B39E520 Suppose that every form $\omega _{u_ t}$ in (11.23.7) is Gauduchon. Then $\int _ X\rho _ t\nu =1$, and $t\mapsto \int _ X\Phi (\rho _ t)\nu $ is nonincreasing for every convex function $\Phi \in C^2((0,+\infty ),\mathbb R)$. In particular,
\[ \| \rho _ t\| _{L^ p(\nu )}\leq \| \rho _ s\| _{L^ p(\nu )} \quad (0\leq s\leq t<T,\ p>1). \]
If $\Phi ''>0$, the derivative of the entropy vanishes at a time $t$ precisely when $\rho _ t=1$, in which case $\dot u_ t=0$.
The hypothesis holds on a Gauduchon surface, and in any dimension it holds if $\mathrm{dd}^{\mathrm{c}}(\omega ^ j)=0$ for $1\leq j\leq n-1$.
Proof.
The derivative of the mass is
\[ \frac{\mathrm{d}}{\mathrm{d}t}\int _ X\rho _ t\nu =\frac nM\int _ X\dot u_ t\mathrm{dd}^{\mathrm{c}}(\omega _{u_ t}^{n-1})=0, \]
and its initial value is one. In (11.23.9) the first term vanishes, while the second is nonpositive, because $\mathrm{d}\rho _ t\wedge \mathrm d^{\mathrm c}\rho _ t$ is a semipositive $(1,1)$-form and $\Phi ''\geq 0$. If $\Phi ''>0$, the vanishing of this term forces $\mathrm{d}\rho _ t=0$, and connectedness together with the mass normalization gives $\rho _ t=1$. The converse and the assertion about $\dot u_ t$ follow from the equation.
On a surface, $\mathrm{dd}^{\mathrm{c}}\omega _{u_ t}=\mathrm{dd}^{\mathrm{c}}\omega =0$. Under the power identities, expand $\omega _{u_ t}^{n-1}$ by the binomial theorem; since the factors $\mathrm{dd}^{\mathrm{c}}u_ t$ are annihilated by $\mathrm d$ and $\mathrm d^{\mathrm c}$, each term has vanishing complex Hessian. These are also the persistence conditions of proposition 11.23.4, theorem 11.1.7.
$\square$
Pome
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