Einstein Metrics and the Yamabe Problem

Which smooth compact n-manifolds admit Riemannian metrics of constant Ricci curvature? A direct variational approach sheds some interesting light on this problem, but by no means answers it. This article surveys some recent results concerning both Einstein metrics and the associated variational problem, with the particular aim of highlighting the striking manner in which the 4-dimensional case differs from the case of dimensions 5. 1 The Total Scalar Curvature For the purposes of this article, the term Einstein metric will mean a Riemannian metric of constant Ricci curvature. In physics jargon, that's to say that an Einstein metric g is a Euclidean-signature solution of the Einstein vacuum equations r = g; for some (unspeciied) value of the cosmological constant 2 R. Here r denotes the Ricci tensor r ab = R c acb of the (positive deenite) Riemannian metric g. Notice that the scalar curvature s = r a a = R ab ab of g is related to the the constant by s = nn; where n is the dimension of the manifold on which g is deened. Einstein metrics are the solutions of a natural variational problem which can be traced back to Hilbert 1]. Choose a smooth compact oriented manifold M of dimension n > 2, and let M = M M denote the space of C 1 Riemannian metrics on M. We may then consider the total scalar curvature functional, or Einstein-Hilbert