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

Using Seiberg-Witten theory, it is shown that any Kähler 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 $H^2(M)=H^+\oplus H^-$. This implies, for example, that any such metric on a minimal ruled surface must be locally symmetric.

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Published 1 January 1995