Abstract
Let g be a hyper-Hermitian metric on a simply connected hypercomplex four-manifold (M,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 explicitely the corresponding hyper-Kähler metrics, which are of cohomegeneity 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.
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Barberis, M.L. Hyper-Kähler Metrics Conformal to Left Invariant Metrics on Four-Dimensional Lie Groups. Mathematical Physics, Analysis and Geometry 6, 1–8 (2003). https://doi.org/10.1023/A:1022448007111
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DOI: https://doi.org/10.1023/A:1022448007111