3.8. Harmonic analysis II: perturbation to Calabi-Yau metric [0441]

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3.8. Harmonic analysis II: perturbation to Calabi-Yau metric

We now shift to the complex geometric viewpoint and solve the complex Monge-Ampère equation by perturbative methods. The main result of the linear theory is (Compare Proposition 2.23):

Proposition 3.32.

Let $-3<\delta<-1$ and $1\ll\nu\ll A^{3/8}$ . Let $f$ be a $T^{2}$ -invariant function compactly supported in $M^{+}_{\nu}$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta}}=1$ . Then there is a $T^{2}$ -invariant function $u$ such that the Poisson equation is approximately solved on $M^{+}_{\nu}$ :

\left\lVert\Delta_{\tilde{g}^{(4)}}u-f\right\rVert_{C^{k,\alpha}_{\delta}}\ll 1,

with the Hessian bound

\left\lVert\nabla^{2}_{\tilde{g}^{(2)}}u\right\rVert_{C^{k,\alpha}_{\delta}}\leq C,\quad\left\lVert du\right\rVert_{C^{k+1,\alpha}_{\delta+1}(M^{+}_{\nu})}\leq CA^{-1/4}.

The constants depend only on $k,\alpha,\delta,\kappa$ and the scale invariant uniform ellipticity constant of $a_{ij}$ .

Proof.

(Sketch) The method is the decomposition and patching argument of Section 2.8, using Proposition 3.22, Lemma 2.20 and Proposition 2.23 as ingredients to provide local parametrices. ∎

Theorem 3.33.

(Ooguri-Vafa type metric on the positive vertex) Fix $k,\alpha,\kappa$ and $0<\epsilon\ll 1$ , and let $1\ll\nu\ll A^{3/8}$ . Then there is a $T^{2}$ -invariant Calabi-Yau metric on $M^{+}_{\nu}$ given by a $T^{2}$ -invariant Kähler potential $\phi^{+}$ ,

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

satisfying the metric deviation estimate

(3.12)

\left\lVert\omega_{+}-\tilde{\omega}^{(4)}\right\rVert_{C^{k,\alpha}_{-1-\epsilon}(M^{+}_{\nu})}\leq C\nu^{2}A^{3/4(-1+\epsilon)},\quad\left\lVert d\phi^{+}\right\rVert_{C^{k+1,\alpha}_{-\epsilon}(M^{+}_{\nu})}\leq C\nu^{2}A^{-1+3\epsilon/4}.

The constants depend only on $k,\alpha,\epsilon,\kappa$ and the scale invariant ellipticity bound on $a_{ij}$ .

Proof.

(Sketch) Given Proposition 3.32, one can set up a Banach iteration scheme to correct the volume form error. A subtlety caused by metric incompleteness is that the parametrix $P_{\nu}$ can only invert sources with compact supports. This problem can be circumvented using the extension norm trick as in Proposition 3.23, and we obtain a Calabi-Yau metric on a shrinked domain $M^{+}_{\nu-1}$ . Changing $\nu$ to $\nu+1$ gives the statement. ∎

Remark 3.9.

The metric lives over a region $M^{+}_{\nu}$ with exponential neck length, but the exponent $\nu$ is not expected to be optimal. For existence results over longer necks one should allow the coupling constants $a_{ij}$ to drift according to the renormalisation flow equation (cf. Section 3.10 and 4.13 for discussions).