Einstein metrics on exotic spheres in dimensions 7, 11, and 15

In a recent article the first three authors proved that in dimension 4m + 1 all homotopy spheres that bound parallelizable manifolds admit Einstein metrics of positive scalar curvature which, in fact, are Sasakian-Einstein. They also conjectured that all such homotopy spheres in dimension 4m−1, m ≥ 2 admit Sasakian-Einstein metrics [Boyer et al. 04], and proved this for the simplest case, namely dimension 7. In this paper we describe computer programs that show that this conjecture is also true for 11-spheres and 15-spheres. Moreover, a program is given that determines the partition of the 8,610 deformation classes of Sasakian-Einstein metrics into the 28 distinct oriented diffomorphism types in dimension 7.