As the observational evidence supporting Einstein's theory of general relativity as the pre-eminent model for the gravitational field in astrophysics and cosmology grows stronger, it becomes increasingly important to use the latest mathematical tools to carefully explore the implications of the theory. This award supports a focus on new procedures for constructing spacetimes representing the the interaction of many bodies, and also on new techniques which allow us to probe the nature of the cosmological singularities predicted by Einstein's theory.
This award also supports work which studies various aspects of Ricci flow, the analytical tool which has proven to be spectacularly successful in proving the Poincare Conjecture and the Geometrization Conjecture in three dimensions. There is much work to do in understanding the details of the dynamics of Ricci flow, and how it might be used to further analyze the relationship between geometry and topology, and this proposal supports such work, including a number of studies of Ricci flow stability.
A graduate student will participate in the supported research as part of a PhD dissertation.
The work supported by this grant involves two categories of research: mathematical relativity and Ricci flow. In this report,I discuss these two categories in order: Mathematical relativity involves the mathematical study of spacetime solutions of Einstein’s equations of general relativity, which describe the gravitational physics of the universe. One portion of my work in this area focuses on the parametrization and construction of initial data sets for such spacetimes. Such data sets must satisfy a set of constraint equations, much like those of electromagnetic theory, but considerably more difficult mathematically. In earlier work, we have been successful in studying initial data sets which satisfy the constraints so long as those set have constant mean curvature, or nearly constant mean curvature. Our focus now is working with initial data sets which do not satisfy these restrictions. In this project, we have managed to parametrize asymptotically flat solutions of the constraints, so long as their shears are small. Much work remains to be done in this direction. Another portion of my mathematical relativity work supported by this grant involves the study of the "strong cosmic censorship" (SCC) conjecture. This conjecture states that among those solutions of the Einstein spacetime field equations which develop singularities of some sort, the generic character of those singularities involves curvature (and therefore tidal force) blowup. SCC has been proven in spacetime solutions with lots of symmetry. Our work seeks to prove it in wider classes of spacetimes. We have made progress in that direction for spacetimes with 2 dimensional isometry groups, and have laid the groundwork for working with spacetimes with 1 dimensional isometry. We have also done work which studies the stability of certain solutions of Einstein’s equations which have accelerated expansion. Since it is believed that this property holds for our universe, understanding the stability of such solutions is important. Ricci flow is a geometrically-based heat type equation which has proven to be very useful in studying the relationship between the topology of a given manifold M and the geometries which are compatible with M. Indeed, Ricci flow was the key ingredient in the recent proof of the Poincare conjecture in 3 dimensions. In studying how Ricci flow works, it is important to understand the properties of the singularities that may develop in a given solution of the Ricci flow equations. One major class of these singularities are those which take the form of a "neckpinch", in which the curvature blows up very locally, in a region which may be modeled at least partially by a cylinder. Our work in this area has first focused on understanding the asymptotic behavior of the geometry of neckpinch singularities which are "degenerate" in the sense that they are partially cylindrical and partially are not. We have shown that there many neckpinch Ricci flow solutions of this kind with very characteristic rates of blowup, and marked by the formation a "Bryant soliton". We understand the axially symmetric case fairly well. We are now studying the case in which the solutions are not axially symmetric. The conjecture is that for many cases, non axially symmetric neckpinches asymptotically approach axially symmetric ones. Ricci flow is the simplest approximation to "Renormalization group flow", which plays an important role in quantum field theory. We have now begun to look at the next order approximation, which we call "RG-2 flow". For the simplest cases, we have shown that some solutions behave much like Ricci flow, while others behave very differently. At least in these simple cases, we can predict which initial geometries lead to Ricci flow type behavior, and which do not.
