2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/160547The 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.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 referencesDifferential GeometryGeneral Relativity and Quantum CosmologyAlgebraic Geometry53C25; 57R57; 83C22text