  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.
