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Annals of PDE

, 3:13 | Cite as

Global Regularity for the 2+1 Dimensional Equivariant Einstein-Wave Map System

  • Lars Andersson
  • Nishanth GudapatiEmail author
  • Jérémie Szeftel
Article

Abstract

In this paper we consider the equivariant 2+1 dimensional Einstein-wave map system and show that if the target satisfies the so called Grillakis condition, then global existence holds. In view of the fact that the 3+1 vacuum Einstein equations with a spacelike translational Killing field reduce to a 2+1 dimensional Einstein-wave map system with target the hyperbolic plane, which in particular satisfies the Grillakis condition, this work proves global existence for the equivariant class of such spacetimes.

Keywords

Einstein’s equations Wave maps 

Notes

Acknowledgements

We are grateful to an anonymous referee, who has done an outstanding and constructive refereeing job, for pointing out an error in an earlier version and for remarks which helped us substantially improve the exposition. L.A. and N.G. thank the Erwin Schödinger Institute, Vienna, for hospitality and support during part of this work. The majority of the work of N.G was supported by the International Max Planck Research School graduate scholarship of Max Planck Society at Albert Einstein Institute, Golm and the DFG Postdoctoral Fellowship GU1513/1-1 at Yale University. N.G is also grateful to Vincent Moncrief for numerous insightful discussions, in particular the experiences he shared from his earlier work on Gowdy spacetimes were influential in some of the approaches adopted for this problem. Further, the authors thank the Mathematical Sciences Research Institute in Berkeley, California, where part of this work was carried out, for hospitality and support. The work carried out during the semester programme on Mathematical General Relativity at MSRI during the fall of 2013 was supported in part by the National Science Foundation under Grant No. 0932078000. The third author is supported by the project ERC 291214 BLOWDISOL.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Lars Andersson
    • 1
  • Nishanth Gudapati
    • 2
    Email author
  • Jérémie Szeftel
    • 3
  1. 1.Max Planck Institute for Gravitational Physics (Albert Einstein Institute)PotsdamGermany
  2. 2.Department of MathematicsYale UniversityNew HavenUSA
  3. 3.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie CurieParisFrance

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