We consider the problem of the Hamiltonian reduction of Einstein's equations on a (3 + 1)-vacuum spacetime that admits a foliation by constant mean curvature (CMC) compact spacelike hypersurfaces M that satisfy certain topological restrictions. After a conformal reduction process, we find that the Einstein flow is described by a dimensionless non-local time-dependent Hamiltonian system ▪ where R − = (−∞, 0) and P reduced = is the reduced phase space. For compact orientable 3-manifolds of Yamabe type −1, we establish the following properties of H reduced: 1. For a hyperbolic manifold M and τ ϵ R − fixed, H reduced has a unique (up to isometry) critical point at the hyperbolic point ( γ , 0) which is a strict local minimum in the non-isometric directions; 2. H reduced is a monotonically decreasing function of t unless γ is hyperbolic and p TT = 0, in which case H reduced is constant in time; 3. for τ ϵ R − fixed, the infimum of H reduced is given by ▪ where σ( M) denotes the σ-constant of M, an important topological invariant of compact manifolds. We conjecture that the local minimum of H reduced is a global minimum, which is equivalent to the conjecture that a hyperbolic metric realizes the σ-constant of a Yamabe type −1 manifold, thereby linking a conjecture of conformal geometry to one of general relativity. We consider applications of these results to the five Bianchi models that compactify to manifolds of Yamabe type −1. For these models, the resulting 3-manifolds are either Seifert fibered spaces, graph manifolds, or hyperbolic manifolds. We show that the reduced Hamiltonian for these models under the Einstein flow asymptotically approaches either the σ-constant or, in the hyperbolic case, what is conjectured to be the σ-constant. In the hyperbolic case, the Einstein flow converges to a hyperbolic metric, whereas in the four non-hyperbolic cases, the Einstein flow collapses the 3-manifold along either circular fibers or embedded tori. Remarkably, in each of these cases of collapse, the collapse occurs with bounded curvature!