TAILIEUCHUNG - Đề tài " Einstein metrics on spheres "

Any sphere S n admits a metric of constant sectional curvature. These canonical metrics are homogeneous and Einstein, that is the Ricci curvature is a constant multiple of the metric. The spheres S 4m+3 , m 1, are known to have another Sp(m + 1)-homogeneous Einstein metric discovered by Jensen [Jen73]. In addition, S 15 has a third Spin(9)-invariant homogeneous Einstein metric discovered by Bourguignon and Karcher [BK78]. In 1982 Ziller proved that these are the only homogeneous Einstein metrics on spheres [Zil82]. . | Annals of Mathematics Einstein metrics on spheres By Charles P. Boyer Krzysztof Galicki and J anos Koll ar Annals of Mathematics 162 2005 557 580 Einstein metrics on spheres By Charles P. BoyER Krzysztof Galicki and JÁNOS Kollar 1. Introduction Any sphere Sn admits a metric of constant sectional curvature. These canonical metrics are homogeneous and Einstein that is the Ricci curvature is a constant multiple of the metric. The spheres S4m 3 m 1 are known to have another Sp m 1 -homogeneous Einstein metric discovered by Jensen Jen73 . In addition S15 has a third Spin 9 -invariant homogeneous Einstein metric discovered by Bourguignon and Karcher BK78 . In 1982 Ziller proved that these are the only homogeneous Einstein metrics on spheres Zil82 . No other Einstein metrics on spheres were known until 1998 when Bohm constructed infinite sequences of nonisometric Einstein metrics of positive scalar curvature on S5 S6 S7 S8 and S9 Boh98 . Bohm s metrics are of cohomogeneity one and they are not only the first inhomogeneous Einstein metrics on spheres but also the first noncanonical Einstein metrics on even-dimensional spheres. Even with Bohm s result Einstein metrics on spheres appeared to be rare. The aim of this paper is to demonstrate that on the contrary at least on odd-dimensional spheres such metrics occur with abundance in every dimension. Just as in the case of Bohm s construction ours are only existence results. However we also answer in the affirmative the long standing open question about the existence of Einstein metrics on exotic spheres. These are differentiable manifolds that are homeomorphic but not diffeomorphic to a standard sphere Sn. Our method proceeds as follows. For a sequence a a1 . am G Z m consider the Brieskorn-Pham singularity Y a Q2 za 0 c Cm and its link L a Y a n S2m-1 1 . i 1 L a is a smooth compact 2m 3 -dimensional manifold. Y a has a natural C -action and L a a natural S 1-action cf. 33 . When the sequence a satisfies certain numerical .

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