In Section 2, we give some background on the Gibbons-Hawking ansatz. We also define the Heisenberg nilmanifolds, and describe the Calabi model space used in the Tian-Yau construction from the Gibbons-Hawking perspective. Lastly, we construct a harmonic function whose associated Gibbons-Hawking ansatz defines the neck region $\mathcal{N}$.

The Calabi model space will be described in Section 3 from a complex geometric perspective, which will be used to obtain the precise asymptotic behavior of the complete hyperkähler Tian-Yau spaces. The main result is Proposition 3.4 which roughly states that a complete Tian-Yau space is exponentially asymptotic to a Calabi model space, up to any arbitrary order of derivatives.

In Section 4, we will establish a Liouville type theorem for harmonic functions which says that any harmonic function of sufficiently small exponential growth on a complete hyperkähler Tian-Yau space has to be a constant. To prove this, the asymptotics proved in Section 3 will be crucial. These will allow us to reduce our problem to a question about harmonic functions on the Calabi model space, which will be solved using separation of variables. Specifically, the Laplace equation on the model space will be reduced to certain linear ODEs, and the quantitative analysis of the harmonic functions reduces to some delicate estimates for Hermite functions. The estimates in Section 4 do not require any advanced theory in hypergeometric functions. Instead the ODE solutions are defined by exponential integrals, which has an advantage that all the calculations in Section 4 are in fact self-contained. This step is already quite involved because it requires us to develop some new elliptic theory on a model space which is a doubly warped product rather than just a cylinder.

Using this work, we will then prove the aforementioned Liouville theorem for half-harmonic 1-forms on a complete hyperkähler Tian-Yau space in Section 5 (see Theorem 5.1). This will later be crucial in the proof of the key uniform estimate (Proposition 9.2) for our gluing construction. An important observation here is that equation (1.28) is equivalent to the $(0,1)$-component of $\phi$ satisfying $\bar{\partial}^{*}\phi^{0,1}=\bar{\partial}\phi^{0,1}=0$ for any choice of compatible integrable complex structure, which will allow us to invoke tools from complex geometry. More precisely, since the equation $\bar{\partial}\phi^{0,1}=0$ depends only on the complex structure, we may study it using a smooth Kähler metric on the compactified Fano manifold. The Kodaira vanishing theorem then ensures that $\phi^{0,1}=\bar{\partial}f$ for some function $f$. Such an $f$ is unique only modulo holomorphic functions, so a priori there is no growth estimate on $f$. However, using the construction of the Tian-Yau metrics and basic elliptic estimates we may find a solution $f$ satisfying some growth estimates. The equation $\bar{\partial}^{*}\phi^{0,1}=0$ then implies that $f$ is harmonic with respect to the Tian-Yau metric. However at this point we are not yet able to conclude using the Liouville theorem for harmonic functions that $f$ must be constant because our growth estimate for $f$ is too weak. We overcome this problem by using separation of variables on the Calabi model space. This final step hinges on the fact that the asymptotic decay rate of the complex structure of the Tian-Yau manifold relative to the Calabi model is much faster than the asymptotic decay rate of the hyperkähler metric.

In Section 6 we will complete the construction of the neck region and construct a closed almost hyperkähler triple on a manifold $\mathcal{M}$. We will also describe some topological invariants of $\mathcal{M}$. Note that at this stage it would be difficult to show directly that $\mathcal{M}$ is actually diffeomorphic to $\K 3$, a fact which will follow easily once we have shown it admits a hyperkähler metric.

Section 7 will focus on the geometry the manifold $\mathcal{M}$. We provide two different methods for analyzing the curvature of the approximate metric. One way is to apply some new types of $\epsilon$-regularity theorems for collapsed Einstein manifolds due to Naber and the fourth author [NZ16], which will yield curvature control once a simple topological condition is verified (see theorem 7.4 in Section 7). On the other hand, in Lemma 7.2 we will also give a direct curvature estimate in the collapsed regions. The remainder of this section will then describe all of the rescaled Gromov-Hausdorff limits for basepoints in each of the various regions.

In Sections 8 and 9, we will set up the weight analysis package and prove the main technical theorems. The first step is to define weighted Hölder spaces which are consistent with the different collapsing behaviors in different regions of $\mathcal{M}$. The remainder of Section 8 will then consist of proving the weighted Schauder estimate in Proposition 8.3. Existence of a bounded right inverse to the linearized operator will be proved in Section 9 (see Proposition 9.4), where we will then complete the proofs of Theorems 1.1 and 1.5.