NEW EXAMPLES OF INHOMOGENEOUS EINSTEIN MANIFOLDS OF POSITIVE SCALAR CURVATURE

The purpose of this note is to announce the explicit construction of a new infinite family of compact inhomogeneous Einstein manifolds of positive scalar curvature in every dimension of the form 4n-5 for n>2. In fact, each manifold has two, non-homothetic, Einstein metrics of positive scalar curvature. Moreover, in every fixed dimension, these families each contain infinitely many distinct homotopy types. Each individual manifold has a Sasakian 3-structure and all of these examples are bi-quotients of unitary groups of the form U(1)\U(n)/U(n-2). In particular, when n=3, we obtain infinite subfamilies of mutually distinct homotopy types where each member of the subfamily is strongly inhomogeneous; that is, these Einstein manifolds are not even homotopy equivalent to any compact Riemannian homogeneous space.