Using twistors, the Gibbons-Hawking ansatz is generalized to investigate 4n-dimensional hyperkähler metrics admitting an action of the n-torus Tn. The hyperkähler case could further admit a tri-holomorphic free action of the n-torus Tn. It turns out that the metric may be written in coordinates adapted to the torus action, in a form similar to the Gibbons-Hawking ansatz in dimension 4, and such that the non-linear Einstein equations reduce to a set of linear equations (essentially saying that certain functions on Euclidean 3-space are harmonic). In the case of 8-manifolds (n = 2), the solutions can be described geometrically, in terms of arrangements of 3-dimensional linear subspaces in Euclidean 6-space.
There are in fact many explicit examples known of metrics on non-compact manifolds with SU(n) or Sp(2n) holonomy. The other holonomy groups automatically yielding Ricci-flat metrics are the special holonomy groups G2 in dimension 7 and Spin(7) in dimension 8. Until fairly recently only three explicit examples of complete metrics (in dimension 7) with G2-holonomy and one explicit example (in dimension 8) with Spin(7)-holonomy were known. The G2-holonomy examples are asymptotically conical and live on the bundle of self-dual two-forms over S4, the bundle of self-dual two-forms over CP2, and the spin bundle of S3 (topologically R4 × S3), respectively. The metrics are of cohomogeneity one with respect to the Lie groups SO(5), SU(3) and SU(2) × SU(2) × SU(2) respectively. A cohomogeneity-one metric has a Lie group acting via isometries, with general (principal) orbits of real codimension one. In particular, if the metric is complete, then X is the holomorphic cotangent bundle of projective n-space T∗CPn, and the metric is the Calabi hyperkähler metric.
The G2-holonomy examples are all examples in which a Lie group G acts with low codimension orbits. This is a general feature of explicit examples of Einstein metrics. The simplest case of such a situation would be when there is a single orbit of a group action, in which case the metric manifold is homogeneous. For metrics on homogeneous manifolds, the Einstein condition may be expressed purely algebraically. Moreover, all homogeneous Ricci-flat manifolds are flat, and so no interesting metrics occur. Then what about cohomogeneity one with respect to G, i.e., the orbits of G are codimension one in general? Here, the Einstein condition reduces to a system of non-linear ordinary differential equations in one variable, namely the parameter on the orbit space. In the Ricci-flat case, Cheeger-Gromoll theorem implies that the manifold has at most one end. In the non-compact case, the orbit space is R+ and there is just one singular orbit. Geometrically, if the principal orbit is of the form G/K, the singular orbit (the bolt) is G/H for some subgroup H ⊃ K; if G is compact, a necessary and sufficient condition for the space to be a smooth manifold is that H/K is diffeomorphic to a sphere. In many cases, this is impossible because of the form of the group G, and so any metric constructed will not be complete.