Christoph Böhm - Inhomogeneous Einstein metrics on spheres

The standard metric on the n-dimensional sphere Sⁿ is Einstein, i.e. its Ricci tensor is a multiple of itself. The first non-trivial example of an Einstein metric on Sⁿ was given by Jensen for n=4m+3 in 1973. Five years later Bourguignon and Karcher described a further Einstein metric on S¹⁵. Up to now these metrics were the only known examples of Einstein metrics on spheres. All of them are homogeneous and Ziller proved that there exists no further homogeneous Einstein metric on Sⁿ. We describe infinitely many inhomogeneous Einstein metrics with positive scalar curvature on S⁵, S⁶, S⁷, S⁸, S⁹. The resulting sequence of Einstein metrics converges to an explicit known limit metric which is smooth outside two submanifolds. Hence we get new examples of manifolds where the Palais-Smale condition for the total scalar curvature functional is not fulfiled.