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

Understand

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].