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Using Seiberg-Witten theory, it is shown that any Kahler metric of constant negative scalar curvature on a compact 4-manifold M minimizes the L 2 -norm of scalar curvature among Riemannian metrics compatible with a fixed decomposition H 2 (M) = H +  H. This implies, for ex- ample, that any such metric on a minimal ruled surface must be locally symmetric.

Polarized 4-Manifolds, Extremal Kähler Metrics, and Seiberg-Witten Theory