FRG: Collaborative Research: Analysis of the Einstein Constraint Equations

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AbstractAward: DMS 1265187, 1262982, 1263431, 1263544PI: Rafe R Mazzeo, Stanford University PI: Michael Holst, University of California - San Diego PI: Jim Isenberg, University of Oregon PI: David Maxell, University of AlaskaThe goal of this project is to understand the extent to which one can parametrize and construct initial data sets for the Einstein evolution equations. We plan to capitalize on the recent progress using the conformal method to obtain new existence results in the nonconstant mean curvature (non-CMC) setting, to understand the limits of these methods, and then to develop alternate techniques toward these same goals, including degree theory, a priori estiimates, and gluing methods. The Lichnerowicz equation, central to the conformal method, is a semilinear elliptic equation. Due to the mixed sign of its nonlinear exponents, it is of a type not yet fully understood. The full Lichnerowicz-Choquet-Bruhat-York set of equations is a more difficult coupled system which incorporates features presenting new analytic subtleties. The ultimate aim is to provide a complete parametrization of initial data sets, particularly in the non-CMC setting, not only on compact backgrounds but also for manifolds with asymptotically Euclidean, hyperbolic or cylindrical ends, all of which are highly relevant for physical applications. In exploring new methods, we plan to use new and advanced analytical tools, as well as increasingly accurate and flexible numerical simulation techniques. Technical advances made in the course of this project should have a substantial application to many other equations of this general type which play important roles in other parts of pure and applied mathematics and mathematical physics. Einstein's gravitational field theory is a remarkably accurate mathematical model of gravitational physics, which does an excellent job of predicting and modeling gravitational phenomena at both the astrophysical and cosmological scales. It is consistent with every known gravitational observation and experiment. From the point of view of underlying mathematics, Einstein's theory involves two very distinct types of equations. The study of dynamics of gravitational fields involves the analysis of the Einstein equations as a nonlinear system of time-dependent evolution partial differential equations (PDE), while the study of initial data sets representing gravitational states involves Riemannian geometry and the study of the Einstein constraint equations as a nonlinear system of time-independent PDE.The last decade has witnessed remarkable progress in under- standing both equations. This project focuses on developing a more complete understanding of the constraint equations. The study of the Einstein equations presents a very important point of contact between mathematics and physics, one which has motivated many advances in differential geometry and PDE on the one side, and which also has provided a compelling and accurate model of the physical world, both on the astrophysical and on the cosmological scales. This project has the potential for settling significant open questions in this area.