  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.
