Abstract: 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.
ab 16:45 Uhr
Arnimallee 3, Raum 006