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

TL;DR

This paper links Einstein-Maxwell equations with extremal Kahler metrics through a Riemannian variational approach and uses Seiberg-Witten theory to explore curvature estimates, advancing understanding in geometric analysis.

Contribution

It introduces a novel variational formulation of Einstein-Maxwell equations and establishes a deep connection with extremal Kahler metrics via Seiberg-Witten theory.

Findings

Reformulation of Einstein-Maxwell equations as a Riemannian variational problem

Demonstration of the relationship between extremal Kahler metrics and Seiberg-Witten theory

Insights into curvature estimates using extremal Kahler metrics

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.