Extremal Kaehler Metrics and Bach-Merkulov Equations

TL;DR

This paper introduces the Bach-Merkulov equations on 4-manifolds, linking extremal Kähler metrics to solutions of these conformally invariant equations and exploring their variational properties.

Contribution

It provides a variational framework for the Bach-Merkulov equations and demonstrates that extremal Kähler metrics are solutions, expanding understanding of conformally invariant geometric PDEs.

Findings

Extremal Kähler metrics are solutions to Bach-Merkulov equations.

Solutions are critical points of the Weyl functional.

On Hirzebruch surfaces, extremal metrics are not always minimizers.

Abstract

In this paper, we study a coupled system of equations on oriented compact 4-manifolds which we call the Bach-Merkulov equations. These equations can be thought of as the conformally invariant version of the classical Einstein-Maxwell equations in general relativity. Inspired by the work of C. LeBrun on Einstein-Maxwell equations on compact Kaehler surfaces, we give a variational characterization of solutions to Bach-Merkulov equations as critical points of the Weyl functional. We also show that extremal Kaehler metrics are solutions to these equations, although, contrary to the Einstein-Maxwell analogue, they are not necessarily minimizers of the Weyl functional. We illustrate this phenomenon by studying the Calabi action on Hirzebruch surfaces.