Hyper-Kähler metrics conformal to left invariant metrics on four-dimensional Lie groups
Let \(g\) be a hyper-Hermitian metric on a simply connected hypercomplex four-manifold \((M,{\mathcal H})\). We show that when the isometry group \(I(M,g)\) contains a subgroup \(G\) acting simply transitively on \(M\) by hypercomplex isometries, then the metric \(g\) is conformal to a hyper-Kähler metric. We describe explicitly the corresponding hyper-Kähler metrics, which are of cohomogeneity one with respect to a 3-dimensional normal subgroup of \(G\). It follows that, in four dimensions, these are the only hyper-Kähler metrics containing a homogeneous metric in its conformal class.
scientific article; zbMATH DE number 1908552