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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Plumbing constructions and the domain of outer communication for 5-dimensional stationary black holes


Authors: Marcus Khuri, Yukio Matsumoto, Gilbert Weinstein and Sumio Yamada
Journal: Trans. Amer. Math. Soc. 372 (2019), 3237-3256
MSC (2010): Primary 53C80, 83C57
DOI: https://doi.org/10.1090/tran/7812
Published electronically: May 30, 2019
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Abstract: The topology of the domain of outer communication for 5-dimensional stationary bi-axisymmetric black holes is classified in terms of disc bundles over the 2-sphere and plumbing constructions. In particular we find an algorithmic bijective correspondence between the plumbing of disc bundles and the rod structure formalism for such spacetimes. Furthermore, we describe a canonical fill-in for the black hole region and cap for the asymptotic region. The resulting compactified domain of outer communication is then shown to be homeomorphic to $ S^4$, a connected sum of $ S^2\times S^2$'s, or a connected sum of complex projective planes $ \mathbb{CP}^2$. Combined with recent existence results, it is shown that all such topological types are realized by vacuum solutions. In addition, our methods treat all possible types of asymptotic ends, including spacetimes which are asymptotically flat, asymptotically Kaluza-Klein, or asymptotically locally Euclidean.


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Additional Information

Marcus Khuri
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794
Email: khuri@math.sunysb.edu

Yukio Matsumoto
Affiliation: Department of Mathematics, Gakushuin University, Tokyo 171-8588, Japan
Email: yukiomat@math.gakushuin.ac.jp

Gilbert Weinstein
Affiliation: Department of Physics and Department of Mathematics, Ariel University, Ariel, 40700, Israel
Email: gilbertw@ariel.ac.il

Sumio Yamada
Affiliation: Department of Mathematics, Gakushuin University, Tokyo 171-8588, Japan
Email: yamada@math.gakushuin.ac.jp

DOI: https://doi.org/10.1090/tran/7812
Received by editor(s): July 13, 2018
Published electronically: May 30, 2019
Additional Notes: The first author acknowledges the support of NSF Grant DMS-1708798.
The fourth author acknowledges the support of JSPS Grants KAKENHI 24340009 and 17H01091.
Article copyright: © Copyright 2019 American Mathematical Society