On the Behavior of Solutions of Einstein's Equations and Other Geometric Partial Differential Equations

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In addition to providing us with an amazingly accurate and beautiful model for studying gravitational physics on both the astrophysical and the cosmological scale, Einstein?s gravitational field theory is a rich source of mathematically interesting questions. A number of the questions we are most interested in pertain to solutions of the Einstein constraint equations. These equations restrict the choice of initial data one can make for an evolving gravitational system. We are interested in parametrizing the set of solutions of the constraints (i.e., finding the degrees of freedom), developing algorithms for constructing solutions, and studying the behavior of solutions of the constraints. We hope to apply the technique of ?gluing? to implement the Einstein constraints. The idea of gluing is that, given a pair of solutions of the constraints, one attempts to connect the two solutions smoothly via a bridge joining a point in each of them. We have applied the gluing technique to the Einstein constraints, and have used it to construct multi-black hole initial data sets, to add wormholes to given solutions, to construct initial data for black holes in cosmological solutions, and to show that any given closed manifold (minus a point) admits an asymptotically flat solution of the constraints. Our work in gluing so far has assumed that the given solutions have a region of constant mean curvature, and are vacuum solutions. We are working to remove these assumptions, which should open up a much wider range of applications.