Let (g,K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold M with non-positive Yamabe invariant (Y(M)). As noted by Fischer and Moncrief, the reduced volume V(k)=(-k/3)^{ 3}Vol_{ g(k)}(M) is monotonically decreasing in the expanding direction and bounded below by V_{ \inf}=(-1/6)Y(M))^{ 3/2}. Inspired by this fact we define the ground state of the manifold M as "the limit" of any sequence of CMC states { (g_{ i},K_{ i})} satisfying: i. k_{ i}=-3, ii. V_{ i} --> V_{ inf}, iii. Q_{ 0}((g_{ i},K_{ i}))< L where Q_{ 0} is the Bel-Robinson energy and L is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of M. 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 (\g,\K)(s)=((-k/3)^{ 2}g,(-k/3))K) where s=-ln(-k) is in [a,\infty). We prove that if E_{ 1}=E_{ 1}((\g,\K))< L (where E_{ 1}=Q_{ 0}+Q_{ 1}, is the sum of the zero and first order Bel-Robinson energies) the flow (\g,\K)(s) persistently geometrizes the three-manifold M and the geometrization is the ground state if V --> V_{ inf}.
Categories: gr-qc