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 
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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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