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

2011/11/21 by Joël Fine, Joel Fine, Fine, Joel · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #math.SG #msc:53C07 #msc:53C21 #msc:53C25

paper · pdf · doi:10.48550/arxiv.1111.5005

49 pages. v2 typos corrected

arxiv created 2011/11/29 · arxiv updated 2011/11/30

Abstract

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 hyperkähler 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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• Twistor geometry of Riemannian 4-manifolds by moving frames

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