Hamiltonian reduction and perturbations of continuously self-similar (n + 1)-dimensional Einstein vacuum spacetimes

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We investigate some of the properties of the vacuum Einstein equations on manifolds of the form V = I × M where M is a compact n-manifold, n ≥ 2, of negative Yamabe type, and where V admits a continuously self-similar solution. One of our primary aims is to relate such solutions on V to critical points of the reduced Hamiltonian Hreduced defined on the reduced phase space Preduced over M. One of our main results is that any continuously self-similar vacuum spacetime which admits a compact constant mean curvature (CMC) slice necessarily induces a negative Einstein metric on that slice and, conversely, that any compact negative Einstein (Riemannian) manifold (M, g) embeds as a CMC slice into a continuously self-similar vacuum spacetime on V. Thus we prove a precise correspondence between the critical points of Hreduced and the set of continuously self-similar CMC sliced (n + 1)-dimensional vacuum spacetimes.Our results are generalizations to n > 3 of some of the results we have previously obtained for n = 2 and 3. In particular, we show that Hreduced is strictly monotonically decreasing in the direction of cosmological expansion except at its critical points where Hreduced is constant and which correspond to continuously self-similar solutions of the Einstein vacuum field equations. A corollary of this result is that discretely self-similar solutions cannot exist in this vacuum spatially compact setting.A new and far reaching generalization of the aforementioned result shows that a family of quasi-local reduced Hamiltonians, defined by integrating the reduced Hamiltonian density over some arbitrary domain D ⊂ M and letting D co-move with the flow also exhibits monotonic decay except for the globally self-similar solutions on which each such quasi-local reduced Hamiltonian is constant. A rather surprising consequence of this analysis is that the quasi-local reduced Hamiltonian on D cannot remain constant, even for a short time interval, unless the spacetime for which it is evaluated is globally self-similar.Finally we extend to arbitrary dimension our earlier result that the infimum of Hreduced taken over the entire reduced phase space determines the topological sigma constant σ(M). For n = 3, our results that Hreduced monotonically decreases along the Einstein flow and has its infimum given directly by σ(M) strongly suggest that the Einstein flow may provide an important new tool in the effort to prove geometrization of 3-manifolds. In fact, a number of compactified Bianchi models are already in hand in which the known or conjectured σ(M) is asymptotically achieved by the monotonic decay of our reduced Hamiltonian.