Annales Henri Poincaré

, Volume 4, Issue 2, pp 369–383 | Cite as

On the Topology of Vacuum Spacetimes

  • J. Isenberg
  • R. Mazzeo
  • D. Pollack
Original article

Abstract.

We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n + 1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold \( \Sigma^n \) to an asymptotically Euclidean solution of the constraints on \( \mathbb{R}^n \). For any \( \Sigma^n \) which does not admit a metric of positive scalar curvature, this provides for the existence of asymptotically flat vacuum spacetimes with no maximal slices. Our main theorem is a special case of a more general gluing construction for nondegenerate solutions of the vacuum constraint equations which have some restrictions on the mean curvature, but for which the mean curvature is not necessarily constant. This generalizes the construction [16], which is restricted to constant mean curvature data.

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Copyright information

© Birkhäuser Verlag Basel, 2003

Authors and Affiliations

  • J. Isenberg
    • 1
  • R. Mazzeo
    • 2
  • D. Pollack
    • 3
  1. 1.University of Oregon, Department of Mathematics, Eugene, OR 97403-1221, USA, e-mail: jim@newton.uoregon.eduUSA
  2. 2.Stanford University, Department of Mathematics, Stanford, CA 94305, USA, e-mail: mazzeo@math.stanford.eduUSA
  3. 3.University of Washington, Mathematics Department, Box 354350, Seattle, WA 98195-4350, USA, e-mail: pollack@math.washington.eduUSA

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