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

2008/03/26 by LeBrun, Claude
#53C25 #57R57 #83C22 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc)

paper · doi:10.48550/arxiv.0803.3734

Abstract

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.

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