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Polarized 4-manifolds, extremal Kähler metrics, and Seiberg-Witten theory. (English) Zbl 0874.53051
Author’s abstract: “ 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”.

MSC:
53C55 Global differential geometry of Hermitian and Kählerian manifolds
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
53C35 Differential geometry of symmetric spaces
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