Extremal Kaehler Metrics and Bach-Merkulov Equations
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.
Keywords
Cite
@article{arxiv.1112.4177,
title = {Extremal Kaehler Metrics and Bach-Merkulov Equations},
author = {Caner Koca},
journal= {arXiv preprint arXiv:1112.4177},
year = {2011}
}