The ground state and the long-time evolution in the CMC Einstein flow

Let (g,K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y (Σ)). As noted by Fischer and Moncrief, the reduced volume V(k) = (−k 3 )V olg(k)(Σ) is monotonically decreasing in the expanding direction and bounded below by Vinf = (−1 6 Y (Σ)) 3 2 . Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(gi,Ki)} satisfying: i. ki = −3, ii. Vi ↓ Vinf , iii. Q0((gi,Ki)) ≤ Λ where Q0 is the Bel-Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally consider a long time and cosmologically normalized flow (g̃, K̃)(σ) = ((−k 3 )g, (−k 3 )K) where σ = −(ln−k) ∈ [a,∞). We prove that if Ẽ1 = E1((g̃, K̃)) ≤ Λ (where E1 = Q0 + Q1, is the sum of the zero and first order Bel-Robinson energies) the flow (g̃, K̃)(σ) persistently geometrizes the three-manifold Σ and the geometrization is the ground state if V ↓ Vinf .