Lemma 16.15.13A7AB2F207A57

Proof. In this proof $C,C_1,C_2,\ldots $ denote constants depending only on $X$, $\omega $, $\kappa $, $\beta $, $C_0$ and $f$. We show that $D_ gS\geq -C_1(S+1)$ and $D_ gH\geq C_0^{-1}S-C_2$, where $H=\operatorname {tr}_ hg$, and apply the maximum principle to $S+KH$ for a large constant $K$.

By lemma 16.15.11(1),(2), $S$ is a smooth function on $X$. The fixed smooth tensors $\widehat R$ (the tensor of lemma 16.15.11(3) for $h$), $\widehat\nabla \widehat R$, $\tau =\operatorname {Ric}(k)+\beta k+\beta t(f_{i\bar j})$ and $\widehat\nabla \tau $ have $h$-norms bounded on $X$ independently of $t\in [0,1]$, because they are smooth on the compact manifold $X$ and depend affinely on $t$; by the assumption on $g$, their $g$-norms are bounded as well.

The Laplacian of $\Psi $. Let $R$ be the tensor of lemma 16.15.11(3) for $g$. Since $\Psi $ has only holomorphic indices,

Put $D_ gT=g^{p\bar q}\nabla _ p\nabla _{\bar q}T$ and $\overline D_ gT=g^{p\bar q}\nabla _{\bar q}\nabla _ pT$ for tensors $T$. On a function $u$ both operators reduce to $D_ gu=g^{p\bar q}\partial _ p\partial _{\bar q}u$, since $\nabla _ p$ adds no term to the antiholomorphic index of $\partial _{\bar q}u$, and $\nabla _{\bar q}$ none to the holomorphic index of $\partial _ pu$. Applying $g^{p\bar q}\nabla _ p$ to the last identity and using $\nabla g^{-1}=0$ and lemma 16.15.11(3), we obtain

By lemma 16.15.11(4) and the first identity of (3), $\overline D_ g\Psi =D_ g\Psi +\mathcal R\Psi $, where

The Laplacian of $S$. By lemma 16.15.11(1) and $\nabla _ r\overline{\Psi ^ l_{pq}}=\overline{\nabla _{\bar r}\Psi ^ l_{pq}}$, which holds because $\nabla _ r$ adds no term to antiholomorphic indices,

The right side is a tensor with one lower holomorphic index $r$, on which $\nabla _{\bar s}$ acts as $\partial _{\bar s}$. Applying lemma 16.15.11(1) once more and contracting with $g^{r\bar s}$, we obtain, with the inner products and norms defined by $g$,

where $\nabla \Psi =(\nabla _ r\Psi ^ m_{ij})$ and $\overline\nabla \Psi =(\nabla _{\bar s}\Psi ^ m_{ij})$; for the last term we used $\overline{g^{r\bar s}}=g^{s\bar r}$. Inserting $\overline D_ g\Psi =D_ g\Psi +\mathcal R\Psi $, we get

in particular $\langle \mathcal R\Psi ,\Psi \rangle _ g$ is real.

Bounds from the equation. In every chart, $\log \det g=\log \det k+\beta (a-tf)$, so

Hence $R_ i{}^ m=g^{m\bar l}\tau _{i\bar l}-\beta \delta ^ m_ i$ is bounded in $g$-norm, and so $|\langle \mathcal R\Psi ,\Psi \rangle _ g|\leq CS$. Moreover $\nabla _ iR_ j{}^ m=g^{m\bar l}\nabla _ i\tau _{j\bar l}$, because $\nabla g^{-1}=0$ and $\nabla \delta =0$. For a tensor the difference $\nabla _ p-\widehat\nabla _ p$ consists of the connection terms with $\Gamma -\widehat\Gamma =\Psi $ in place of $\Gamma $; thus $\nabla _ i\tau _{j\bar l}=\widehat\nabla _ i\tau _{j\bar l}-\Psi ^ a_{ij}\tau _{a\bar l}$ and

By (16.15.14) it follows that $|D_ g\Psi |_ g\leq C(1+S^{1/2})$. Discarding the two nonnegative gradient terms in (16.15.15), we conclude that

A compensating term. We use the trace $H=\operatorname {tr}_ hg$. Let $x\in X$, and take the coordinates of lemma 16.15.4 at $x$. By that lemma, identity (16.15.5) holds at $x$, with $|E|\leq C_ EH\operatorname {tr}_ gh\leq C$ by the assumption on $g$, and $D_ h\log \det g=D_ h\log \det k+\beta (H-\operatorname {tr}_ hk-tD_ hf)$ is bounded by $C$. At $x$ the coefficients $\widehat\Gamma $ vanish, so $\Psi ^ m_{ij}=\lambda _ m^{-1}\partial _ ig_{j\bar m}$ and

Hence the quadratic sum in (16.15.5) is at least $(\min _ p\lambda _ p)S\geq C_0^{-1}S$, and $D_ gH\geq C_0^{-1}S-C_2$ at $x$. Both sides are functions on $X$, so this inequality holds everywhere.

The maximum principle. Choose $K$ with $KC_0^{-1}\geq C_1+1$, and let $x_0$ be a maximum point of $S+KH$. There the complex Hessian of $S+KH$ is negative semidefinite, so

Thus $S\leq C_1+KC_2$ at $x_0$. Since $H\leq nC_0$ on $X$, the function $S+KH$, and hence $S$, is bounded on $X$.

First derivatives of $g$. In a chart of the fixed atlas, $\partial _ pg_{i\bar q}=\widehat\nabla _ pg_{i\bar q}+\widehat\Gamma ^ l_{pi}g_{l\bar q}=g_{m\bar q}\Psi ^ m_{pi}+\widehat\Gamma ^ l_{pi}g_{l\bar q}$ by lemma 16.15.11(2). On a compact subset of the chart, $h$ lies between two positive multiples of the identity matrix and $\widehat\Gamma $ is bounded, so both terms are bounded, by the bounds for $S$ and $g$. Finally $\partial _{\bar p}g_{i\bar q}=\overline{\partial _ pg_{q\bar i}}$, so all first derivatives of the $g_{i\bar j}$ in the underlying real coordinates are bounded. $\square$