I n t r o d u c t i on A Riemannian manifold (M, g) is Einstein if its Ricci tensor satisfies Ric(g) = ~ 9 g for some constant ~. The terminology results from the fact that if (M, g) is a Lorentz 4-manifold, then the Einstein condition is precisely Einstein's field equation in vacuo. In this paper, we are interested in the situation where (M, g) admits a compact Lie group action by isometries with cohomogeneity one, i.e., an action whose principal orbits are hypersurfaces in M. In the study of Einstein metrics, cohomogeneity one examples are of particular interest because the regular part of (M, g) corresponds in a natural way to a spatially homogeneous Lorentz Einstein manifold. Indeed, the first non-K~ihler inhomogeneous compact Riemannian Einstein manifold was constructed by the physicist D. Page from the Taub-NUT solution [18]. His construction was generalized by Berard-Bergery in the unpublished preprint [3] and later independently by Page and Pope [19]. Subsequently, many authors have studied Einstein metrics of cohomogeneity one in various specific instances. While these works are too numerous to list, we mention especially the work of Sakane, for he constructed in [22] the first non-homogeneous examples of K~ihler-Einstein
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