On the global evolution problem in 2+1 gravity

Details On the global evolution problem in 2+1 gravity On the global evolution problem in 2+1 gravity

1996-10-10

arxiv.org/abs/gr-qc/9610013v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

14 pages, amsart

Scientific paper

10.1016/S0393-0440(97)87804-7

Existence of global CMC foliations of constant curvature 3-dimensional maximal globally hyperbolic Lorentzian manifolds, containing a constant mean curvature hypersurface with $\genus(\Sigma) > 1$ is proved. Constant curvature 3-dimensional Lorentzian manifolds can be viewed as solutions to the 2+1 vacuum Einstein equations with a cosmological constant. The proof is based on the reduction of the corresponding Hamiltonian system in constant mean curvature gauge to a time dependent Hamiltonian system on the cotangent bundle of Teichm\"uller space. Estimates of the Dirichlet energy of the induced metric play an essential role in the proof.

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