Model geometries in the space of Riemannian structures and Hamilton's flow

Abstract

Hamilton's theorem (1986) states that, under suitable conditions, a given (three- or four-dimensional) Riemannian manifold can be smoothly deformed, via a heat-type equation, into a space of constant sectional curvature. This result, already used by the authors in discussing Ellis' fitting problem (1987) in relativistic cosmology, is here examined in detail. On using some recently proven compactness properties of the space of Riemannian structures, they provide a natural setting for understanding the geometrical rationale behind Hamiltonian's results. This allows us to simplify the existing proof of the global nature of Hamilton's initial-value problem and to discuss it as a distinguished dynamical system on the space of Riemannian metrics. Finally, they briefly argue about the possible applications of these results either to general relativity or to the quantum field theory of extended objects.