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