Einstein Metrics on Spheres

Any sphere S admits a metric of constant sectional curvature. These canonical metrics are homogeneous and Einstein, that is the Ricci curvature is a constant multiple of the metric. The spheres S, m > 1 are known to have another Sp(m + 1)-homogeneous Einstein metric discovered by Jensen [Jen73]. In addition, S has a third Spin(9)-invariant homogeneous Einstein metric discovered by Bourguignon and Karcher [BK78]. In 1982 Ziller proved that these are the only homogeneous Einstein metrics on spheres [Zil82]. No other Einstein metrics on spheres were known until 1998 when Böhm constructed infinite sequences of nonisometric Einstein metrics, of positive scalar curvature, on S, S, S, S, and S [Böh98]. Böhm’s metrics are of cohomogeneity one and they are not only the first inhomogeneous Einstein metrics on spheres but also the first non-canonical Einstein metrics on even-dimensional spheres. Even with Böhm’s result, Einstein metrics on spheres appeared to be rare. The aim of this paper is to demonstrate that on the contrary, at least on odddimensional spheres, such metrics occur with abundance in every dimension. Just as in the case of Böhm’s construction, ours are only existence results. However, we also answer in the affirmative the long standing open question about the existence of Einstein metrics on exotic spheres. These are differentiable manifolds that are homeomorphic but not diffeomorphic to a standard sphere S. Our method proceeds as follows. For a sequence a = (a1, . . . , am) ∈ Z m + consider the Brieskorn–Pham singularity