Gravitational solitons and complete Ricci flat Riemannian manifolds of infinite topological type

Abstract

We present several new space-periodic solutions of the static vacuum Einstein equations in higher dimensions, both with and without black holes, having Kasner asymptotics. These latter solutions are referred to as gravitational solitons. Further partially compactified solutions are also obtained by taking appropriate quotients, and the topologies are computed explicitly in terms of connected sums of products of spheres. In addition, it is shown that there is a correspondence, via Wick rotation, between the spacelike slices of the solitons and black hole solutions in one dimension less. As a corollary, the solitons give rise to complete Ricci flat Riemannian manifolds of infinite topological type and generic holonomy, in dimensions $4$ and higher.