A gauge theoretic approach to the anti-self-dual Einstein equations

Details A gauge theoretic approach to the anti-self-dual Einstein equations A gauge theoretic approach to the anti-self-dual Einstein equations

2011-11-21

arxiv.org/abs/1111.5005v2

Mathematics

Differential Geometry

49 pages. v2 typos corrected

Scientific paper

In [29], Plebanski reformulated the anti-self-dual Einstein equations with non-zero scalar curvature as a first order PDE for a connection in an SO(3)-bundle over the four-manifold. The aim of this article is to place this differential equation in a new framework, in which it is both elliptic and a stationary point of a parabolic flow. To do this, we exploit a link with definite connections (introduced in [12]) to draw an analogy with instantons and the Yang-Mills flow. This picture leads to a natural conjecture, analogous to one made by Donaldson concerning hyperk\"ahler 4-manifolds [9]. It also provides a moment-map description of the anti-self-dual Einstein equations with non-zero scalar curvature.

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