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Polarized 4-Manifolds, Extremal Kähler Metrics, and S-W Theory

Claude LeBrun

dg-gaarXiv:dg-ga/9506002

Abstract

Using Seiberg-Witten theory, it is shown that any Kaehler metric of constant negative scalar curvature on a compact 4-manifold M minimizes the L2-norm of scalar curvature among Riemannian metrics compatible with a fixed decomposition H2(M)=(H+) + (H-). This implies, for example, that any such metric on a minimal ruled surface must be locally symmetric.

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