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14.14.6 The third derivatives and the torsion term.D5E11D27FAAA

We now estimate the third derivatives in this inequality. Define

\begin{gather*} E=\sum _{k,p} f_ k(|v_{pk}|^2+|v_{p\bar k}|^2),\qquad S=\sum _{k,p} f_ k(|v_{pk}|+|v_{p\bar k}|),\\ \mathcal C=-D^2\Phi _ g(\nabla _1g,(\nabla _1g)^*)\geq 0. \end{gather*}

The first differentiated equation is

BE0A6E37B652 14.14.7
\begin{equation} \sum _ k f_ k\nabla _ p g_{k\bar k}=h_ p. \end{equation}

Since $P_\gamma (W)=Z$ and the connection preserves $\gamma $, one has the exact contractions

5D951DE4F0FD 14.14.8
\begin{equation} \sum _ k f_ k\nabla _ p W_{k\bar k}=\sum _ i t_ i\nabla _ p Z_{i\bar i}, \qquad \sum _ k f_ k\nabla _{\bar1}\nabla _1W_{k\bar k} =\sum _ i t_ i\nabla _{\bar1}\nabla _1Z_{i\bar i}. \end{equation}

The weights in these identities are evaluated at the point; they are not differentiated.

The large weights are controlled as follows. By (14.13.6), the only weight $t_ i$ not bounded by $(n-1)f_1$ is $t_1$. In $Z_{1\bar1}$, every gradient index differs from $1$, by (14.13.4). Thus a pure second derivative in $\nabla _ pZ_{1\bar1}$ has an index $q>1$, and it can be weighted by $f_ q$, which is comparable to $\mathcal F$; here we use that pure derivatives commute in these coordinates. A mixed second derivative can similarly be placed in column $q$ by conjugate symmetry. All other $t_ i$ are bounded by a constant times every $f_ k$, and derivatives of coefficients cost $C\sqrt K\mathcal F$. Applying these observations to the first identity in (14.14.8), and then to (14.14.7), we obtain, for each $p$,

EACA66A12E82 14.14.9
\begin{equation} \left|\sum _ k f_ kv_{k\bar k p}\right| +\left|\sum _ k f_ kv_{k\bar k\bar p}\right| \leq C(S+\sqrt K\mathcal F). \end{equation}

The connection terms in the differentiated equation contribute weighted mixed second derivatives and terms bounded by $C\sqrt K\mathcal F$, which are already on the right. The bounded derivatives of $h$ are absorbed using the positive lower bound for $\mathcal F$ in lemma 14.13.5.

Next we insert the second derivative of the equation into the eigenvalue estimate. Since partial derivatives of $v$ commute,

\[ g_{1\bar1,k\bar k} =g_{k\bar k,1\bar1}+W_{1\bar1,k\bar k} -W_{k\bar k,1\bar1}+O(1). \]

Differentiating the equation twice covariantly gives

\[ \sum _ k f_ k\nabla _{\bar1}\nabla _1g_{k\bar k} =h_{1\bar1}+\mathcal C. \]

The conversion to partial derivatives uses

\[ \nabla _{\bar1}\nabla _1g_{k\bar k} =g_{k\bar k,1\bar1} -2\operatorname {Re}\sum _ q\overline{\Gamma ^ q_{1k}}g_{k\bar q,1} +O(\lambda ). \]

We rewrite the derivative of $g$ in this term using $g_{k\bar q,1}=g_{1\bar q,k}+W_{k\bar q,1}-W_{1\bar q,k}+O(1)$ and its conjugate. The first part is absorbed by the off-diagonal squares in (14.14.3), at the cost of $C\lambda \mathcal F+C\sum _ k f_ k|g_{1\bar1,k}|$. The part involving $W_{1\bar q,k}$ is bounded by $C\lambda \mathcal F+C\sum _{k,p}f_ k|v_{pk}|$. For the remaining part we use

\[ 2\operatorname {Re}\sum _ q\overline{\Gamma ^ q_{1k}}W_{k\bar q,1} =W_{k\bar k,1\bar1} -\nabla _{\bar1}\nabla _1W_{k\bar k}+O(\sqrt K). \]

This cancels the partial second derivative $-W_{k\bar k,1\bar1}$ above, and we obtain

9.40
\begin{align} \sum _ k f_ k\widetilde\lambda _{k\bar k} \geq {}& \mathcal C+\sum _ k f_ k (W_{1\bar1,k\bar k}-\nabla _{\bar1}\nabla _1W_{k\bar k})\notag \\ & -C\left(\sum _ k f_ k|g_{1\bar1,k}| +\sum _{k,p}f_ k|v_{pk}|+\lambda \mathcal F\right). \tag{14.14.10} \end{align}

The second contraction in (14.14.8) bounds the negative torsion term here. In the part with weight $t_1$, the possible third derivatives are $v_{q1\bar1}$ and their conjugates with $q>1$; all other parts have weights bounded by $Cf_1$. We commute partial derivatives and use

\[ v_{1\bar1,k}=g_{1\bar1,k}-\chi _{1\bar1,k}-W_{1\bar1,k}. \]

The exclusions in lemma 14.13.2 ensure that any remaining pure derivative $v_{11}$ carries only the small weight $Cf_1$, and derivatives of the connection contribute second derivatives with the same weighted bounds. It follows that

\[ \left|\sum _ k f_ k\nabla _{\bar1}\nabla _1W_{k\bar k}\right| \leq C\left(\sum _ k f_ k|g_{1\bar1,k}| +\sum _{k,p}f_ k|v_{pk}|+\lambda \mathcal F\right). \]

For the positive torsion term, write $W_{1\bar1}=\sum _ p(c^ pv_ p+\overline{c^ p}v_{\bar p})$ with fixed smooth coefficients $c^ p$. Two derivatives give $c^ p\sum _ k f_ kv_{pk\bar k}$ and its conjugate, plus terms bounded by $C(S+\sqrt K\mathcal F)$. The former are bounded by (14.14.9), and in $S$ the mixed second derivatives can be bounded by $C\lambda $, giving

\[ \left|\sum _ k f_ k W_{1\bar1,k\bar k}\right| \leq C\left(\sum _{k,p}f_ k|v_{pk}|+\lambda \mathcal F\right). \]

Substitute these estimates in (14.14.10), multiply (14.14.5) by $f_ k$ and sum over $k$. The third derivatives in the latter are bounded by (14.14.9) and $\phi '|\partial v|\leq C/\sqrt K$, and the resulting linear terms in the second derivatives are absorbed by a small part of $E/K$, at the cost of $C\mathcal F$. Since $\widetilde H$ has a local maximum, $\sum _ kf_ k\widetilde H_{k\bar k}\leq 0$, and thus

35CF4803162E 14.14.11
\begin{equation} \begin{split} 0\geq {}& \frac{\mathcal C}{\lambda } -\sum _ k f_ k\frac{|a_ k|^2}{\lambda ^2} +\frac{E}{6K}+\phi ''\sum _ k f_ k|V_ k|^2 +\psi ''\sum _ k f_ k|v_ k|^2+\psi '\sum _ k f_ kv_{k\bar k}\\ & -C\left(\sum _ k f_ k\frac{|g_{1\bar1,k}|}{\lambda }+\mathcal F\right). \end{split} \end{equation}
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