Rigidity in vacuum under conformal symmetry
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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.
KeywordsLorentzian rigidity Vacuum equations Conformal symmetry
Mathematics Subject Classification53C50 83C75
GJG’s research was supported in part by NSF Grants DMS-1313724 and DMS-1710808.
- 9.Galloway, G.J.: Some rigidity results for spatially closed space-times. Mathematics of gravitation, Part I (Warsaw, 1996), Banach Center Publications, vol. 41, pp. 21–34. Polish Academy of Science, Warsaw (1997)Google Scholar
- 16.O’Neill, B.: Semi-Riemannian Geometry. Pure and Applied Mathematics, vol. 103. Academic Press Inc., New York (1983)Google Scholar
- 17.Yano, K.: The Theory of Lie Derivatives and Its Applications. Bibliotheca Mathematica, vol. 3. North-Holland Pub. Co., Amsterdam (1957)Google Scholar
- 18.Yau, S.-T.: Problem Section. Annals of Mathematics Studies, No. 102, pp. 669–706. Princeton University Press, Princeton (1982)Google Scholar