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Christoph Böhm - Inhomogeneous Einstein metrics on spheres



CHRISTOPH BÖHM, Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario  L8S 4K1, Canada
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.


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Next: Jeffrey Boland - Magnetic Up: 1)  Differential Geometry and Global Previous: Igor Belegradek - Pinching
 

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