A gravitational instanton is a complete non-compact hyperKähler 4-manifold with $\int|\text{Rm}|^{2}d\text{Vol}<\infty$. The theory of gravitational instantons is very rich, with important contributions from Kronheimer, Atiyah, Hitchin, Hein, and many others. Recent breakthrough made by Chen and Chen [1] is a decisive step towards a complete classification. A conspicuous feature of this classification program is the crucial role played by the volume growth rate. In the Euclidean volume growth rate case, these are the ALE metrics (‘asymptotically locally Euclidean’) classified by Kronheimer. In the sub-Euclidean volume growth case, in all known situations the asymptotic geometry near infinity is approximately a flat torus fibration over a flat base.

We will not attempt to review this extensive literature, but limit ourselves to examine a simple class of examples known as multi-Taub-NUT metrics. In the Gibbons-Hawking coordinates (cf. Section 1.2), this is given by the potential

$$V=A+\sum_{i=1}^{k}\frac{1}{2\sqrt{|\mu-\mu_{i}|^{2}+|\eta-\eta_{i}|^{2}}},$$

where $(\mu_{i},\eta_{i})$ are disjoint given points on the base $\mathbb{R}^{3}=\mathbb{R}_{\mu}\oplus\mathbb{C}_{\eta}$, and $k\geq 1$. The asymptotic geometry is given by a degree $k$ circle bundle over the complement of a compact region in $\mathbb{R}^{3}$, whose circle fibres have approximate length $2\pi A^{-1/2}$. This behaviour is known as asymptotically locally flat, or ALF for short. The $k=1$ case is the usual Taub-NUT metric.

Let’s assume for convenience that the $\eta_{i}$ are all distinct, which is the generic situation. From the holomorphic perspective, the multi-Taub-NUT metrics lives on the smooth algebraic varieties

$$\{z_{0}z_{1}=F(\eta)=(\eta-\eta_{1})(\eta-\eta_{2})\ldots(\eta-\eta_{k})\}\subset\mathbb{C}^{3}_{z_{0},z_{1},\eta},$$

with nowhere vanishing holomorphic volume form $\Omega=\frac{1}{F^{\prime}(\eta)}\sqrt{-1}dz_{0}\wedge dz_{1}$, and the circle action is

$$e^{i\theta}\cdot(z_{0},z_{1})=(e^{-i\theta}z_{0},e^{i\theta}z_{1}),$$

which ensures $\iota_{\frac{\partial}{\partial\theta}}\Omega=d\eta$.

A crucial aspect of multi-Taub-NUT metrics is that they come in a moduli space, determined by the positions of $(\mu_{i},\eta_{i})$. In general, the fact that a family of geometric objects has natural moduli indicates the possibility that in some degenerate limit they decompose into more primary objects, and the parameters in the moduli comes from the parameters in these building blocks and the combinatorics of the gluing construction. This is the case when the spatial separation distance of the monopole points $(\mu_{i},\eta_{i})\in\mathbb{R}^{3}$ is far larger than the circle length parameter $A^{-1/2}$. Then we can view the multi-Taub-NUT metric as obtained from gluing $k$ copies of the Taub-NUT metrics, whose curvature centres are far separated and therefore whose mutual interaction is weak.

Now we can try to push this story to higher dimensions. The natural generalisation of complete hyperKähler 4-folds is complete Calabi-Yau manifolds. Since in our Taub-NUT type $\mathbb{C}^{3}$ example the Riemannian curvature does not decay at infinity along $\mathfrak{D}_{i}$, the total $L^{2}$-curvature integral is infinite. Finding the correct generalised notion of finite curvature condition is clearly fundamental to any classification program. We do not fully understand what this notion is. A tentative idea compatible with Chen and Chen’s work [1] and our Taub-NUT type $\mathbb{C}^{3}$ example is to require that $|\text{Rm}|\leq C$ globally and $|\text{Rm}|(x)\leq C\text{dist}(x,0)^{-2-\epsilon}$ for some $\epsilon>0$ in the generic region.

In the direction of constructing more examples, we comment that Hein’s existence package is by no means limited to the case of $\mathbb{C}^{n}$. Focusing on complex dimension 3, the distinguished role of our Taub-NUT type metrics on $\mathbb{C}^{3}$ is instead that they are more primary objects, and in particular ought to have a more rigid moduli space, than most of the other 3-dimensional complete Calabi-Yau metrics with similar behaviours. It is perhaps best to illustrate this by a conjectural example which generalises the multi-Taub-NUT metrics.

Let $\eta_{i}$ be all distinct and take the smooth algebraic varieties

$$\{z_{0}z_{1}z_{2}=F(\eta)=(\eta-\eta_{1})(\eta-\eta_{2})\ldots(\eta-\eta_{k})\}\subset\mathbb{C}^{4}_{z_{0},z_{1},z_{2},\eta},$$

with nowhere vanishing holomorphic volume form $\Omega=\frac{-1}{F^{\prime}(\eta)}dz_{0}\wedge dz_{1}\wedge dz_{2}$. These admit a $T^{2}$-action

$$e^{i\theta_{1}}\cdot(z_{0},z_{1},z_{2})=(e^{-i\theta_{1}}z_{0},e^{i\theta_{1}}z_{1},z_{2}),\quad e^{i\theta_{2}}\cdot(z_{0},z_{1},z_{2})=(e^{-i\theta_{2}}z_{0},z_{1},e^{i\theta_{2}}z_{2})$$

which ensures $\Omega(\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}})=d\eta$. It seems likely that Hein’s package can be made to provide a multi-parameter family of complete Calabi-Yau metrics on these varieties. Morever, when the $T^{2}$-fibres have much smaller lengths compared to the spatial separation of $\eta_{i}$, then the author expects such metrics to have a gluing description in terms of our Taub-NUT type metric on $\mathbb{C}^{3}$. On the other extreme, if we allow $\eta_{i}$ to collide, then we may see new metric behaviours not yet understood in the literature.