Proof.
The quantity $\mathcal R$ below is the squared norm of the difference between the Chern connection of $g$ and the flat connection. We derive a differential inequality for $\mathcal R$ in which the torsion enters through fixed derivatives of $\vartheta $, and combine it with the trace in a maximum principle; every constant is kept a power of $M$.
We compare the flat connection of $\beta $ with the Chern connection $\nabla $ of $g$. Write $g^{k\bar\ell }$ for the coefficients of the inverse matrix and $\bar\nabla $ for covariant differentiation in $(0,1)$ directions; tensor norms and inner products without subscript are computed with $g$. The difference of the two connections and its squared norm are
\[ \Psi ^ k_{ij}=g^{k\bar\ell }\partial _ i g_{j\bar\ell }, \qquad \mathcal R=|\Psi |_ g^2. \]
The torsion of $\nabla $ is determined by the fixed background, because the third derivatives of $u$ cancel in it. Its lowered and raised components are
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15.18.38
\begin{equation} C_{ij\bar\ell } =\partial _ i\vartheta _{j\bar\ell }-\partial _ j\vartheta _{i\bar\ell }, \qquad \mathcal T^ k_{ij}=g^{k\bar\ell }C_{ij\bar\ell }. \end{equation}
Consequently, for a fixed integer $N$,
14.39
\begin{align} |\mathcal T|& \leq CM^ N,\nonumber \\ |\nabla \mathcal T|+|\bar\nabla \mathcal T| & \leq CM^ N(1+\sqrt{\mathcal R}),\tag{15.18.39}\\ |\nabla ^2\mathcal T|+|\nabla \bar\nabla \mathcal T|+|\bar\nabla ^2\mathcal T| & \leq CM^ N(1+\mathcal R+|\nabla \Psi |+|\bar\nabla \Psi |).\nonumber \end{align}
Every factor here is either a fixed derivative of $\vartheta $, a connection factor $\Psi $, or a factor of $g^{\pm 1}$; this explains the polynomial dependence.
For tensor fields, use $\Delta _ g=g^{p\bar q}\nabla _ p\nabla _{\bar q}$. We need the following curvature identities. With
\[ \operatorname {Rm}^ k{}_{i\bar j l}=-\partial _{\bar j}\Psi ^ k_{il}, \qquad \operatorname {Rm}_{i\bar j l\bar k}=g_{m\bar k}\operatorname {Rm}^ m{}_{i\bar j l}, \]
set
\[ \operatorname {Ric}^{(2)}_{l\bar k}=g^{i\bar j}\operatorname {Rm}_{i\bar j l\bar k}, \qquad \operatorname {Ric}^{(1)}_{l\bar k}=g^{i\bar j}\operatorname {Rm}_{l\bar k i\bar j} =-\partial _ l\partial _{\bar k}\log F. \]
The two curvature interchanges are
\begin{align*} \operatorname {Rm}_{i\bar j l\bar k}-\operatorname {Rm}_{l\bar j i\bar k} & =-\nabla _{\bar j}C_{il\bar k},\\ \operatorname {Rm}_{i\bar j l\bar k}-\operatorname {Rm}_{i\bar k l\bar j} & =-\nabla _ i\overline{C}_{\bar j\bar k l}. \end{align*}
They follow by differentiating the torsion identity and its conjugate. Successive interchange and contraction give
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15.18.40
\begin{equation} \operatorname {Ric}^{(2)}_{l\bar k} =\operatorname {Ric}^{(1)}_{l\bar k} -g^{i\bar j}\bigl ( \nabla _ l\overline C_{\bar j\bar k i} +\nabla _{\bar j}C_{il\bar k} \bigr ). \end{equation}
Metric compatibility, (15.18.39), and the fixed smooth formula for $\operatorname {Ric}^{(1)}$ imply
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15.18.41
\begin{equation} |\operatorname {Ric}^{(2)}|\leq CM^ N(1+\sqrt{\mathcal R}), \qquad |\nabla \operatorname {Ric}^{(2)}| \leq CM^ N(1+\mathcal R+|\nabla \Psi |+|\bar\nabla \Psi |). \end{equation}
Moreover,
\[ \nabla _{\bar q}\Psi ^ k_{ij}=-\operatorname {Rm}^ k{}_{i\bar q j}, \]
and the Chern–Bianchi identity, including its torsion term, is
\[ \nabla _ p\operatorname {Rm}^ k{}_{i\bar q j}-\nabla _ i\operatorname {Rm}^ k{}_{p\bar q j} =-\mathcal T^ r_{pi}\operatorname {Rm}^ k{}_{r\bar q j}. \]
Consequently,
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15.18.42
\begin{equation} \Delta _ g\Psi ^ k_{ij} =-\nabla _ i(\operatorname {Ric}^{(2)})^ k{}_ j +g^{p\bar q}\mathcal T^ r_{pi}\operatorname {Rm}^ k{}_{r\bar q j}. \end{equation}
If $A^ k{}_ l=(\operatorname {Ric}^{(2)})^ k{}_ l$, define its action on a tensor of type $(1,2)$ by
\[ (A\mathbin {\cdot }\Psi )^ k_{ij} =A^ k{}_ l\Psi ^ l_{ij}-A^ l{}_ i\Psi ^ k_{lj}-A^ l{}_ j\Psi ^ k_{il}. \]
Differentiating the squared norm and using metric compatibility gives
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15.18.43
\begin{equation} \Delta _ g\mathcal R =|\nabla \Psi |^2+|\bar\nabla \Psi |^2 +2\operatorname {Re}\langle \Delta _ g\Psi ,\Psi \rangle -\operatorname {Re}\langle A\mathbin {\cdot }\Psi ,\Psi \rangle . \end{equation}
There is no mixed torsion term because the Chern torsion has type $(2,0)$. Using (15.18.39), (15.18.41), and (15.18.42) in (15.18.43), Young's inequality absorbs the products of $\sqrt{\mathcal R}$ with $|\nabla \Psi |+|\bar\nabla \Psi |$. We obtain
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\begin{equation} \Delta _ g\mathcal R\geq -CM^ N(\mathcal R^{3/2}+1). \end{equation}
Let $s=\operatorname {tr}_\beta g$. Direct differentiation as in (15.18.10) gives
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\begin{equation} \Delta _ gs\geq M^{-1}\mathcal R-CM, \qquad |\partial s|_ g^2\leq nM^2\mathcal R. \end{equation}
Take a cutoff $\zeta $ supported in $B$, equal to one on $B_{1/2}$, set $\eta =\zeta ^2$ and $h=2nM-s$, so $nM\leq h\leq 2nM$, and maximize
\[ Q=\frac{\eta \mathcal R}{h}. \]
At a positive interior maximum of $Q$, the identity $\partial \mathcal R/\mathcal R=\partial h/h-\partial \eta /\eta $ gives
\[ \begin{split} 0\geq \frac{\Delta _ gQ}{Q} & =\frac{\Delta _ g\mathcal R}{\mathcal R} +\frac{\Delta _ g\eta }{\eta }-\frac{\Delta _ gh}{h} -\frac{2|\partial \eta |_ g^2}{\eta ^2} +\frac{2\operatorname {Re}\langle \partial \eta ,\partial h\rangle _ g}{\eta h}. \end{split} \]
We have $|\partial \eta |_ g^2\leq CM\eta $, $|\Delta _ g\eta |\leq CM$ and $|\partial h|_ g\leq \sqrt n M\sqrt{\mathcal R}$. Insert (15.18.44) and (15.18.45) and multiply by $\eta Q$. The positive term is $M^{-1}Q^2$. The two terms involving $\sqrt{\mathcal R}$ are bounded in absolute value by a fixed power of $M$ times $Q^{3/2}$, since $\mathcal R=hQ/\eta $ and $h\leq 2nM$. The remaining cutoff terms are bounded by $CM^ N(Q+1)$. Therefore
\[ 0\geq M^{-1}Q^2-CM^ N(Q^{3/2}+Q+1). \]
Thus $Q$, and then $\mathcal R$ on $B_{1/2}$, are bounded by a fixed polynomial in $M$. Since $|\partial g|_\beta ^2\leq M^3\mathcal R$, this proves (15.18.36).
If $s\leq L$, the determinant lower bound gives $\lambda _{\min }(g)\geq (\inf F)L^{1-n}$, so $M$ is a fixed power of $1+L$. For a ball $B_ r(p)$, set $u_ r(z)=r^{-2}u(p+rz)$ and use the coefficient matrices $\vartheta _ r(z)=\vartheta (p+rz)$, $g_ r(z)=g(p+rz)$ and $F_ r(z)=F(p+rz)$. Then $g_ r=\vartheta _ r+\mathrm{dd}^{\mathrm{c}}u_ r$ in the unit coordinates and $\det g_ r=F_ r$. For $r\leq 1$ the derivatives of the fixed data remain uniformly bounded. The unit-ball estimate for the ordinary coefficient derivatives $\partial g_ r$ and their conjugates gives the factor $r^{-1}$ in (15.18.37).
$\square$
Pome
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