Pome

Lemma 8.4.5174BDA603BF2

Lemma 8.4.5.174BDA603BF2 Put $s=|\partial u|_ g^2$ and $K=1+\sup _ Xs$. For every $\eta >0$ one has

\[ Ls\geq \sum _{i,k}F_ i(|u_{ki}|^2+|u_{k\bar i}|^2) +2\operatorname {Re}\sum _ k f_ k u_{\bar k}-\eta Q-C_\eta K\mathcal F. \]

The constant depends only on the metric and $\eta $.

Proof. Differentiate the squared norm using metric compatibility. Its quadratic part is the displayed sum. Commuting the remaining third derivatives gives

\[ u_{ki\bar i}=u_{i\bar i;k}-T^ p_{ik}u_{p\bar i} +R^ p_{i\bar i k}u_ p, \qquad u_{\bar k i\bar i}=u_{i\bar i;\bar k} -\overline{T^ p_{ik}}u_{i\bar p}. \]

The first differentiated equation is $\sum _ iF_ i u_{i\bar i;k}=f_ k$, because the background $g$ is parallel. At the diagonal point the torsion errors are bounded by $C\sqrt K\sum _ iF_ i(|\lambda _ i|+1)$, and the curvature errors by $CK\mathcal F$. Use

\[ C\sqrt K F_ i|\lambda _ i| \leq \eta ' F_ i\lambda _ i^2+C_{\eta '}KF_ i \]

with a sufficiently small $\eta '$ for the finite sum. This gives the asserted bound. $\square$

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