4.10. Harmonic analysis III: perturbation to Calabi-Yau metric [046V]

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4.10. Harmonic analysis III: perturbation to Calabi-Yau metric

We now shift to the complex geometric perspective and solve the complex Monge-Ampère equation by perturbative methods. Below is main result of the linear theory, which is parallel to Proposition 2.23 and 3.32. The idea is to patch together as in Proposition 3.32 the local parametrices provided by Proposition 4.31 and 4.36.

Proposition 4.37.

Given $-3<\delta<-1$ and $1\ll\nu\ll A^{3/8}$ , let $f$ be an $S^{1}$ -invariant function compactly supported in $M^{-}_{\nu}$ with norm $\left\lVert f\right\rVert_{C^{\alpha}_{\delta}}=1$ . Then there is an $S^{1}$ -invariant function $u$ approximately solving the Poisson equation:

\left\lVert\Delta_{g^{(2)}}u-f\right\rVert_{C^{\alpha}_{\delta}(M^{-}_{\nu})}\ll 1,

with the Hessian bound

\left\lVert\nabla^{2}_{g^{(2)}}u\right\rVert_{C^{\alpha}_{\delta}(M^{-}_{\nu})}\leq C,\quad\left\lVert du\right\rVert_{C^{1,\alpha}_{\delta+1}(M^{-}_{\nu})}\leq CA^{-1/2}.

The constants depend only on $\delta,\alpha,\kappa$ and the scale invariant ellipticity bound on $a_{p\bar{q}}$ .

Combined with the initial error estimate (4.31) this allows us to set up a Banach iteration scheme to perturb $\omega^{(2)}$ to a Calabi-Yau metric, parallel to Theorem 3.33. This involves shrinking domain from $M^{-}_{\nu}$ to $M^{-}_{\nu-1}$ and changing $\nu$ to $\nu+1$ .

Theorem 4.38.

(Ooguri-Vafa type metric on the negative vertex) Fix $k,\alpha,\kappa$ and $0<\epsilon\ll 1$ , and let $1\ll\nu\ll A^{3/8}$ . Then there is an $S^{1}$ -invariant Calabi-Yau metric on $M^{-}_{\nu}$ with $S^{1}$ -invariant Kähler potential $\phi^{-}$ ,

\omega_{-}=\omega^{(2)}+\sqrt{-1}\partial\bar{\partial}\phi^{-},\quad\omega_{-}^{3}=\frac{3}{4}\sqrt{-1}\Omega\wedge\overline{\Omega},

with metric deviation estimate $\left\lVert\nabla^{2}_{g^{(2)}}\phi^{-}\right\rVert_{C^{\alpha}_{-1-\epsilon}(\mathcal{B}^{-}_{\nu})}\leq C\nu^{2}A^{-3/4}.$ The constants depend only on $\alpha,\epsilon,\kappa$ and the scale invariant ellipticity bound on $a_{p\bar{q}}$ .