, Volume 142, Issue 2, pp 435-450

Rigidity of Einstein 4-manifolds with positive curvature

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An Einstein metric with positive scalar curvature on a 4-manifold is said to be normalized if Ric=1. A basic problem in Riemannian geometry is to classify Einstein 4-manifolds with positive sectional curvature in the category of either topology, diffeomorphism, or isometry. It is shown in this paper that if the sectional curvature K of a normalized Einstein 4-manifold M satisfies the lower bound K≥ε0, ε0≡( -23)/120≈0.102843, or condition (b) of Theorem 1.1, then it is isometric to either S 4, RP 4 with constant sectional curvature K=1/3, or CP 2 with the normalized Fubini-Study metric. As a consequence, both the normalized moduli spaces of Einstein metrics which satisfy either one of the above two conditions on S 4 and CP 2 contain only a single point. In particular, if M is a smooth 4-manifold which is homeomorphic to either S 4, RP 4, or CP 2 but not diffeomorphic to any of the three manifolds, then it can not support any normalized Einstein metric which satisfies either one of the conditions.

Oblatum 4-II-1999 & 4-V-2000¶Published online: 16 August 2000