Pure and Applied Mathematics Quarterly

Volume 20 (2024)

Number 4

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

Pages: 1895 – 1921

DOI: https://dx.doi.org/10.4310/PAMQ.2024.v20.n4.a12

Authors

Marcus Khuri (Department of Mathematics, Stony Brook University, Stony Brook, NY, USA)

Martin Reiris (Centro de Matemática, Universidad de la República, Montevideo, Uruguay)

Gilbert Weinstein (Department of Mathematics & Department of Physics, Ariel University, Ariel, Israel)

Sumio Yamada (Department of Mathematics, Gakushuin University, Tokyo, Japan)

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.

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This paper is dedicated to Demetrios Christodoulou. The work herein is inspired by the problem that he suggested to the third author, for his PhD thesis, almost four decades ago

Received 17 April 2022

Received revised 24 August 2022

Accepted 19 October 2022

Published 18 July 2024