Towards Finding the Second Best Einstein Metric in Low Dimensions

Abstract

In the following work we investigate the structure of Einstein manifolds with positive scalar curvature whose curvature operator is sufficiently close to the identity operator in dimensions below 12. An Einstein manifold with positive scalar curvature, that is not locally isometric to the round sphere, is called the second best Einstein manifold if its curvature operator minimizes the angle to the identity operator among all of these manifolds. In dimensions above 11 the search for the second best Einstein manifold turns out to be of purely algebraic nature and is, by a conjecture of B\"ohm and Wilking, locally isometric to the product of spheres of a certain dimension. In lower dimensions the angle of the curvature operator of the complex projective plane to the identity operator is an obstruction for proving the same result with algebraic methods. We show that, assuming a conjecture of B\"ohm and Wilking, there exists an angle slightly smaller than the angle of the product of spheres to the identity operator such that any simply connected Einstein manifold with positive scalar curvature whose angle of the curvature operator to the identity is smaller than this angle is isometric to the round sphere. We will also be able to compute this angle explicitely.