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.
Streaming audio, video and slides
Streaming
audio and slides
video (Real
Player, ~ 15 Mo)
audio (Real
Player, ~ 8 Mo)
slides (pdf)
slides (ps)
slides (jpg.tar)
slides (tex)
First lecture
-
-
-
-
Second lecture
-
-
-
-
Third lecture
-
-
-
-
All three lectures
-
-
-
-
-
-
-
Copyright 29.VIII.02 by P.T.Chrusciel, A.Chopin, and V. Moncrief