On the global evolution problem in 2+1 gravity

Existence of global constant mean curvature (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 CMC gauge to a time-dependent Hamiltonian system on the cotangent bundle of Teichmuller space. Estimates of the Dirichlet energy of the induced metric play an essential role in the proof.