A gauge theoretic approach to Einstein 4-manifolds
The aim of the paper is the reformulation of Einstein's equations in dimension \(4\) in the language of gauge theory. The paper wants to describe constructions from physics in a language accessible to mathematicians. For a definite connection over a \(4\)-manifold, a metric is defined using the curvature of the connection. The connection can be viewed as the potential for the metric. To the connection, an action is associated (the volume of the metric) whose critical points are Einstein metrics. Topological bounds are also investigated and are related with the anti-self-dual property of the Einstein metric. The given reformulation of Einstein's equations gives a link between Einstein \(4\)-manifolds and other types of special manifolds. The formalism also gives a new variational approach to Einstein metrics, because the traditional action principle is not suitable for the calculus of variations.
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