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The Nonvacuum Einstein Flow on Surfaces of Negative Curvature and Nonlinear Stability

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Abstract

We prove future nonlinear stability for the Einstein–Vlasov system in 2+1 dimensions, for a manifold of the type \({\Sigma \times \mathbb{R}}\), where \({\Sigma}\) is closed with genus > 1, in the expanding direction. This is the first stability result for the Einstein–Vlasov system for the case of vanishing cosmological constant without symmetry assumptions in the cosmological case. As an essential part of the proof, we introduce a method to obtain decay for the distribution function by the construction of geometric Sobolev-type norms arising from the Sasaki metric on the tangent bundle of spacelike hypersurfaces. Those norms are modified by the implementation of a correction mechanism yielding energy estimates with optimal decay properties of the relevant norms.

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Authors and Affiliations

  1. Gravitational Physics, Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090, Vienna, Austria

    David Fajman

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  1. David Fajman
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Correspondence to David Fajman.

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Communicated by P. T. Chruściel

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Fajman, D. The Nonvacuum Einstein Flow on Surfaces of Negative Curvature and Nonlinear Stability. Commun. Math. Phys. 353, 905–961 (2017). https://doi.org/10.1007/s00220-017-2842-9

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