Proposition 12.9.15.9D81484FC945 Fix $0<\delta \leq \Lambda $. For every smooth real $v$ with
\[ \delta \omega \leq \Omega _ v:=\omega +\mathrm{dd}^{\mathrm{c}}v\leq \Lambda \omega , \]
let $\rho _ v$ be its Gauduchon factor normalized by
\[ \mathrm{dd}^{\mathrm{c}}(\rho _ v\Omega _ v^{n-1})=0, \qquad \int _ X\rho _ v\, \mathrm{d}V=1. \]
There is $C_ H\geq 1$, depending only on $X,\omega ,\mathrm{d}V,\delta ,\Lambda $, such that $C_ H^{-1}\leq \rho _ v\leq C_ H$ on $X$ for every such $v$.
Proof.
Existence and uniqueness of $\rho _ v$ follow from theorem 11.4.2. Fix a real reference metric on $X$, and put $Q_ v=\Omega _ v^{n-1}$ and $J_ v=\Omega _ v^ n/\mathrm{d}V$. Since $d\Omega _ v=d\omega $,
82DFE47055F1
12.9.16
\begin{equation} dQ_ v=(n-1)d\omega \wedge \Omega _ v^{n-2}. \end{equation}
Both $Q_ v$ and $dQ_ v$ have uniformly bounded coefficients in fixed charts.
Define a real tensor $A_ v$ and vector field $b_ v$ by
\[ \langle A_ vd\phi ,d\psi \rangle \, \mathrm{d}V =n\, d\psi \wedge \mathrm d^{\mathrm c}\phi \wedge Q_ v, \qquad (b_ v\cdot d\phi )\, \mathrm{d}V =-n\, \mathrm d^{\mathrm c}\phi \wedge dQ_ v. \]
Only the $(1,1)$ part of $d\psi \wedge \mathrm d^{\mathrm c}\phi $ contributes to the first identity; it is
\[ \frac{\mathrm i}{4\pi } (\partial \psi \wedge \bar\partial \phi +\partial \phi \wedge \bar\partial \psi ). \]
Thus $A_ v$ is symmetric and uniformly positive definite, with a uniform upper bound. The second identity and (12.9.16) give a uniform bound for $b_ v$.
For $L_{\Omega _ v}\phi =n\mathrm{dd}^{\mathrm{c}}\phi \wedge Q_ v/\Omega _ v^ n$, Stokes' theorem gives
\begin{align*} \int _ X\psi J_ vL_{\Omega _ v}\phi \, \mathrm{d}V & =-n\int _ Xd\psi \wedge \mathrm d^{\mathrm c}\phi \wedge Q_ v +n\int _ X\psi \mathrm d^{\mathrm c}\phi \wedge dQ_ v\\ & =-\int _ X\langle A_ vd\phi ,d\psi \rangle \, \mathrm{d}V -\int _ X\psi b_ v\cdot d\phi \, \mathrm{d}V. \end{align*}
In particular, the adjoint equation for $\rho _ v$ is
\[ \operatorname {div}_{\mathrm{d}V}(A_ vd\rho _ v+b_ v\rho _ v)=0. \]
In each coordinate chart, multiplying the coefficients by the fixed smooth volume density converts this to Euclidean divergence form while preserving uniform ellipticity and a uniform bound for the vector coefficient. Lemma 12.9.12 applies. A fixed finite cover by inner cubes has a connected overlap graph, since $X$ is connected. Chaining its local estimates gives $\sup _ X\rho _ v\leq C_ H\inf _ X\rho _ v$. The normalization and $\mathrm{d}V(X)=1$ imply the claimed two-sided bound.
$\square$
Pome
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