The analytic step is aimed at perturbing the first order ansatz into a genuine Calabi-Yau metric, and the techniques involved overlap substantially in all three cases.

A central issue, roughly put, is to produce a parametrix for the right inverse to the Laplacian with accurate control on weighted Hölder norm estimates. Some of the main difficulties are:

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  The first order corrected metric is multiscaled, namely it has very different characteristic behaviours in different regions and at different length scales.

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  The initial error decays slowly.

The core idea in our methodology is divide and conquer. We decompose the source function according to its support. The contribution supported sufficiently away from the discriminant locus is inverted approximately using the Euclidean Green operator, reflecting the fact that the constant solution is the zeroth order approximation to the metric ansatz. Afterwards the source function is effectively supported near the discriminant locus. We then use a Green operator adapted to the Taub-NUT fibration near the discriminant locus to cure the remaining source.

We now turn to specifics. The 3 cases are arranged in pedagogical order, and each case contains most difficulties of previous cases. As a general policy, detailed proofs will be omitted if the main techniques appeared previously.

In the Taub-NUT type $\mathbb{C}^{3}$ case, the parametrix is used to improve the approximation to the Calabi-Yau condition asymptotically outside a compact region. Once the decay of the approximation error is sufficiently fast, we can appeal to a non-compact version of Yau’s solution to the Calabi conjecture, developed in H-J. Hein’s thesis [12], to turn the ansatz into a genuine Calabi-Yau metric with effective estimates.

Here a difficulty caused by the slow decay of error is that the inverse of the Laplacian is not well behaved in the weighted Hölder spaces. Instead it is preferable to work with the zeroth order operator $\nabla^{2}\Delta^{-1}$, which controls how to correct a Kähler metric for a given amount of volume form error. The advantage is that this operator maps between function spaces with the same Hölder weights, the operator norm is not affected by rescaling the metric, and crucially the Schwartz kernel has two extra order of decay compared to $\Delta^{-1}$.

In the positive vertex case, the main new difficulty is to prove exponential decay of higher Fourier modes. This comes down to mapping properties of the periodic Euclidean Green operator, ultimately thanks to the exponential decay of the higher Fourier modes of the periodic Newtonian potential.

The second new difficulty is that that the volume form error does not decay, and in fact grows logarithmically at large distance, causing problem for perturbation theory over an exponentially long region. The strategy is to first correct the error inside the generic region in the generalised Gibbons-Hawking framework, using the periodic Green operator. We then switch to the complex geometric viewpoint and solve the complex Monge-Ampère equation perturbatively, which avoids the difficulty of the generalised Gibbons-Hawking equation near the discriminant locus.

The third new difficulty comes from metric incompleteness: the Laplacian has no good mapping property in the naïve weighted Hölder spaces. In our approach, this means the parametrix is only defined on compactly supported sources, but the outputs are generally not compactly supported. A formal trick called extension norms [27] effectively allows us to assume the source is compactly supported. This circumvents the need to impose a non-canonical boundary condition.

In the negative vertex case, the main new difficulty comes from the curved nature of the discriminant locus $S$, making it harder to produce a parametrix near $S$. A closely related issue is that there is no obvious a priori choice of smooth topology such that the first order metric ansatz is smooth along $S$. These problems force us to work in weighted Hölder spaces with low regularity, in which it makes no sense to speak of an arbitrarily high order of differentiability. Crucially there is enough regularity to make the Laplacian well defined. The smooth topology emerges a posteriori only after solving the complex Monge-Ampère equation. The solution itself defines a complex structure, hence induces a smooth topology, and the compatibility of the metric with this smooth topology is a consequence of the well known regularity theory for complex Monge-Ampère equation.