The Einstein-Maxwell Equations, Extremal Kahler Metrics, and Seiberg-Witten Theory

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LeBrun, Claude
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Abstract
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The Einstein-Maxwell equations on a smooth compact 4-manifold are reformulated as a purely Riemannian variational problem analogous to Calabi's variational problem for extremal Kahler metrics. Next, Seiberg-Witten theory is used to show that these two problems are in fact intimately related. Extremal Kahler metrics are then used to probe the limits of Seiberg-Witten curvature estimates. The article then concludes with a brief survey of some recent results on extremal Kahler metrics.
Comment: 21 pages, LaTeX2e. To appear in "The Many Facets of Geometry: a Tribute to Nigel Hitchin." Final version includes two new results, as well as many added references
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Mathematics - Differential Geometry, General Relativity and Quantum Cosmology, Mathematics - Algebraic Geometry, 53C25, 57R57, 83C22
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