Weyl curvature, Einstein metrics, and Seiberg-Witten theory

We show that solutions of the Seiberg-Witten equations lead to non-trivial estimates for the $L^{2}$-norm of the Weyl curvature of a compact Riemannian 4-manifold. These estimates are then used to derive new obstructions to the existence of Einstein metrics on smooth compact 4-manifolds with a non-zero Seiberg-Witten invariant. These results considerably refine those previously obtained \cite{lno} by using scalar-curvature estimates alone.

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Published 1 January 1998