V. Moncrief

Title: The Einstein flow and geometrization

Lecture 1: The Cauchy Problem and 3-Manifold Topology
Abstract: We study some aspects of the Cauchy problem for Einstein's equations and how they relate to the possible topologies of the spatial manifold. The Thurston conjectures for 3-manifold topologies are briefly described and some of their implications for the global behavior of solutions of Einstein's equations are discussed.

Lecture 2: Hamiltonian Reduction and the Einstein Flow
Abstract: We discuss, especially for spatial manifolds of the negative Yamabe type, the Hamiltonian reduction of Einstein's equations and describe some of the known properties of the reduced Hamiltonian. In particular, the infimum of this Hamiltonian is shown to determine the topological invariant of the spatial manifold known as the sigma-constant. A quasi-local variant of the reduced Hamiltonian is also discussed.

Lecture 3: Einstein Spaces as Attractors for Einstein's Equations
Abstract: Einstein's equations in n + 1 dimensions are reformulated (after a trivial rescaling) as an autonomous dynamical system on a suitable reduced phase space. The fixed points of this system are identified and shown to be determined by the negative Einstein metrics (if any exist) of the spatial manifold. If a certain stability property holds (for which there are currently no known counterexamples) these fixed points are shown to be attractors for nearby solutions in the direction of cosmological expansion.

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Copyright 29.VIII.02 by P.T.Chrusciel, A.Chopin, and V. Moncrief