Rigidity in vacuum under conformal symmetry

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Abstract

Motivated in part by Eardley et al. (Commun Math Phys 106(1):137–158, 1986), in this note we obtain a rigidity result for globally hyperbolic vacuum spacetimes in arbitrary dimension that admit a timelike conformal Killing vector field. Specifically, we show that if M is a Ricci flat, timelike geodesically complete spacetime with compact Cauchy surfaces that admits a timelike conformal Killing field X, then M must split as a metric product, and X must be Killing. This gives a partial proof of the Bartnik splitting conjecture in the vacuum setting.

Keywords

Lorentzian rigidity Vacuum equations Conformal symmetry 

Mathematics Subject Classification

53C50 83C75 

Notes

Acknowledgements

GJG’s research was supported in part by NSF Grants DMS-1313724 and DMS-1710808.

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MiamiCoral GablesUSA
  2. 2.Department of MathematicsBinghamton University SUNYBinghamtonUSA

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