5.3 Assembling the pieces [0249]

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5.3 Assembling the pieces

Theorem 5.1.

(Existence of complete Calabi-Yau metric) There is a potential $\phi_{rel}$ such that $\phi=\phi_{glue}+\phi_{rel}$ solves the complex Monge-Ampère equation with decay bound

(dd^{c}\phi)^{n}=K_{0}\sqrt{-1}^{n^{2}}\Omega\wedge\overline{\Omega},\quad\left\lVert\phi_{rel}\right\rVert_{k,\alpha,loc}=O(\tilde{\rho}^{-q}),

where $q$ can be chosen as any positive number smaller than $\frac{2n-4}{n+2}$ .

Proof.

We have verified the weighted Sobolev inequality (cf. section 5.2), the existence of the distance-like function (cf. section 4.7), and the existence of $C^{k,\alpha}$ quasi-atlas (cf. section 4.1, 4.2).

The volume form error for $dd^{c}\phi_{glue}$ decays like $O(\tilde{\rho}^{-\frac{2n(2n-1)}{(n-1)(n+2)}})$ (cf. (36)). On the other hand, the volume growth rate is $O(\tilde{\rho}^{p^{\prime}})$ with $2<p^{\prime}=\frac{4n}{n+2}<\frac{2n(2n-1)}{(n-1)(n+2)}$ . In particular the volume error is bounded by $O(\tilde{\rho}^{-p^{\prime}})$ . Applying Hein’s package, we can find a $C^{k,\alpha}$ bounded solution $\phi_{rel}$ , such that

(dd^{c}\phi_{glue}+dd^{c}\phi_{rel})^{n}=(1+Err_{2})^{-1}(dd^{c}\phi_{glue})^{n}=K_{0}\sqrt{-1}^{n^{2}}\Omega\wedge\overline{\Omega},

with decay estimate $|\phi_{rel}|=O(\tilde{\rho}^{2-p_{0}+\epsilon})=O(\tilde{\rho}^{-q})$ . Since the charts on the local universal covers have harmonic radius scale $O(\rho)$ , elliptic regularity improves the decay estimate to local $C^{k,\alpha}$ -norms. ∎

The smallness of $dd^{c}\phi_{rel}$ near infinity means that the main features of the asymptotic geometry are preserved. In particular, the volume growth of the Calabi-Yau metric $dd^{c}\phi$ is $\text{Vol}(B(\tilde{\rho}))\sim\tilde{\rho}^{\frac{4n}{n+2}}$ . The tangent cone at infinity refers to the pointed Gromov-Hausdorff limit of rescaled geodesic balls centred at a fixed reference point. In our case, the rescaling procedure obliviates the $T^{2}$ and $Y$ -fibres . The tangent cone is topologically $\mathbb{R}_{\geq 0}^{2}$ with the variables $x_{1},x_{2}$ , and metrically it is up to a constant the Hessian metric

g_{\infty}=\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}dx_{i}dx_{j},

where $u(x_{1},x_{2})$ is the solution to the non-archimedean Monge-Ampère equation. The renormalized measure on the tangent cone, which comes from pushing forward the complex Monge-Ampère measure, is up to constant factor $dx_{1}dx_{2}$ .