There is currently tremendous interest in geometric PDE, due in
part to the geometric flow program used successfully to attack the
Poincare and Geometrization Conjectures. Geometric PDE also play
a primary role in general relativity, where the (constrained) Einstein
evolution equations describe the propagation of gravitational waves
generated by collisions of massive objects such as black holes.
The need to validate this geometric PDE model of gravity has led to
the recent construction of (very expensive) gravitational wave
detectors, such as the NSF-funded LIGO project. In this lecture, we
consider the non-dynamical subset of the Einstein equations called
the Einstein constraints; this coupled nonlinear elliptic system must
be solved numerically to produce initial data for gravitational wave
simulations, and to enforce the constraints during dynamical simulations,
as needed for LIGO and other gravitational wave modeling efforts.

The Einstein constraint equations have been studied intensively for
half a century; our focus in this lecture is on a thirty-year-old open
question involving existence of solutions to the constraint equations
on space-like hyper-surfaces with arbitrarily prescribed mean extrinsic
curvature. All known existence results have involved assuming either
constant (CMC) or nearly-constant (near-CMC) mean extrinsic curvature.
After giving a survey of known CMC and near-CMC results through 2007,
we outline a new topological fixed-point framework that is fundamentally
free of both CMC and near-CMC conditions, resting on the construction of
"global barriers" for the Hamiltonian constraint. We then present
such a barrier construction for case of closed manifolds with positive
Yamabe metrics, giving the first known existence results for arbitrarily
prescribed mean extrinsic curvature. Our results are developed in the
setting of a ``weak'' background metric, which requires building up a
set of preliminary results on general Sobolev classes and elliptic
operators on manifold with weak metrics. However, this allows us
to recover the recent ``rough'' CMC existence results of Choquet-Bruhat
(2004) and of Maxwell (2004-2006) as two distinct limiting cases of our
non-CMC results. Our non-CMC results also extend to other cases such
as compact manifolds with boundary.

Time permitting, we also outline some new abstract approximation theory
results using the weak solution theory framework for the constraints; an
application of which gives a convergence proof for adaptive finite
element methods applied to the Hamiltonian constraint.

This is joint work with Gabriel Nagy and Gantumur Tsogtgerel.