Scalar Curvature, Metric Degenerations, and the Static Vacuum Einstein Equations on 3-manifolds, Ii

Consider the problem of finding a constant curvature metric on a given closed 3-manifoldM . It is known that essential spheres and tori, (except in the case of flat manifolds), prevent the existence of such metrics, but it is unknown if there are other topological obstructions. We approach this issue by recasting it as a natural variational problem on the space of metrics M onM . The problem then becomes two-fold. First, find a natural curvature functional I on M whose minima are necessarily constant curvature metrics. Second, try to detect topological information on M from the geometric behavior of minimizing sequences for I. First, which curvature functional I should one choose that is best suited for these tasks? In dimension 3, the full curvature tensor R is determined algebraically by the Ricci curvature r, and so functionals involving R or r have essentially the same properties. However, one does not expect to be able to detect the existence of essential spheres inM from the behavior of minimizing sequences for ∫ |r|2dV for example, or from the geometric structure of limits of minimizing sequences. Choosing other Lp norms of r either does not help or leads to severe analytic problems. These difficulties are explained in detail in [An3, §7]; we also point out that there exist critical points of ∫ |r|2dV on the space of unit volume metrics M1 which are not constant curvature, c.f. [La]. This leads one to consider functionals involving the scalar curvature s. (It would also be interesting to consider the Chern-Simons functional in this respect, but we will not do so here). The Einstein-Hilbert action S was studied in some detail in this regard in the predecessor paper [AnI]. Here, we focus on functionals related to the L2 norm of s, i.e. S = (v ∫ sdV ), (0.1)