Self-Dual Solutions to Euclidean Gravity

The discovery of self-dual instanton solutions in Euclidean Yang-Mills theory [I] has recently stimulated a great deal of interest in self-dual solutions to Einstein’s theory of gravitation. One would expect that the relevant instanton-like metrics would be those whose gravitational fields are self-dual, localized in Euclidean spacetime and free of singularities. In fact, solutions have been found which have the additional interesting property that the metric approaches a flat metric at infinity. These solutions are called “asymptotically locally Euclidean” metrics because, in spite of their asymptotically flat local character, their global topology at infinity differs from that of ordinary Euclidean space. Since the Yang-Mills instanton potential approaches a pure gauge at infinity, this class of Einstein solutions closely resembles the Yang-Mills case. The first examples of asymptotically locally Euclidean metrics were the self-dual solutions given by the authors in Ref. [2]. Belinskii, Gibbons, Page and Pope [3] then studied the general class of self-dual Euclidean Bianchi type IX metrics and showed that only metric II of Ref. [2] could describe a nonsingular manifold. Gibbons