Extremal Kaehler Metrics and Bach-Merkulov Equations

Caner Koca

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arXiv · 1112.4177

Extremal Kaehler Metrics and Bach-Merkulov Equations

Caner Koca

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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.

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BibTeX RIS

Caner Koca. 2011-12-18. Extremal Kaehler Metrics and Bach-Merkulov Equations. https://arxiv.org/abs/1112.4177

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