On the Behavior of Solutions of Einstein's Equations and Solutions of Geometric Heat Flow Systems

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Einstein's gravitational field theory provides a beautiful and remarkably accurate means for modeling gravitational physics on both the astrophysical and cosmological scales. It can be used to predict the observational consequences (both in terms of electromagnetic and gravitational radiation) of black holes and neutron stars and ordinary stars colliding, and it can also be used to discern from observations what the Big Bang was like. One particularly useful way to work with Einstein's theory is by formulating it as an initial value problem: According to this formulation, to construct spacetimes of use in modeling gravitational physics, one first chooses an initial state for the spacetime (at some arbitrary time of interest), and one then constructs the spacetime by evolving both into the past and the future of this initial state. Einstein's equations (at the heart of Einstein's theory) both control possible choices of the initial state, and determine how the evolution proceeds. The research supported in this proposal involves how to make choices of the initial state which satisfy the Einstein constraint equations; it also involves determining the generic behavior of solutions as one approaches "singular regions" of the spacetime (near the Big Bang, for example). Besides studies of the behavior of solutions of Einstein's equations, this grant also supports studies of solutions of geometric heat flow equations such as the Ricci flow and mean curvature flow. Here, the interest is in the mathematical relationship between topological spaces and the types of curvature that they can support. Remarkably, some of the techniques used in the study of geometric heat flow solutions are also useful in studying solutions of Einstein's equations.Among the specific projects supported by this grant are the following:1) Solutions of the Einstein constraint equations: Two approaches have been developed for constructing and studying solutions of the constraints: The first of these, the conformal method, works beautifully for constant mean curvature ("CMC") and near-CMC solutions of the vacuum or electrovac Einstein constraints (with nonpositive cosmological constant), but appears to have major problems otherwise. This grant supports work which studies these problems---non-existence and non-uniqueness of solutions---in a number of cases, including asymptotically Euclidean ("AE") and asymptotically hyperbolic ("AH") solutions, as well as solutions on closed manifolds. The second approach, gluing, allows known solutions of the constraints to be joined to produce new ones--e.g., N-body initial data sets. AH initial data must be "shear-free" if it is to be used to produce asymptotically flat spacetimes; hence this grant supports work to develop gluing techniques which allow the joining at infinity of a pair of shear-free AH solutions, thereby producing a new shear-free AH solution (with a single asymptotic region).2) Strong Cosmic Censorship: For almost 50 years, one of the major questions in mathematical relativity has been if the ubiquitous geodesic incompleteness in maximal spacetime developments predicted by the Hawking-Penrose "singularity theorems" is generically accompanied by spacetime curvature blowup. Geodesically incomplete solutions of Einstein's equations with bounded curvature (allowing extensions across a Cauchy horizon) are known; but the "Strong Cosmic Censorship ("SCC") conjecture suggests that this does not happen generically. Model versions of SCC have been proven for families of solutions, such as the Gowdy spacetimes. In these proofs, verifying "AVTD" behavior (dominance of time derivatives over space derivatives near the singular region) has been a crucial tool. The PI and collaborators has developed the singular initial value problem as a way of identifying AVTD behavior, and proposes to use it to find non-analytic AVTD solutions among vacuum solutions with one Killing field, and among Einstein-scalar solutions with no Killing fields. Other supported works seeks to show that the AVTD behavior of Kasner solutions is stable among solutions with two Killing fields. 3) Expanding Cosmologies: The PI and his collaborators propose to use a combination of numerical and analytical studies to explore the expanding direction of model cosmological spacetimes. There is good evidence for strongly attracting "entropic" behavior. This grant supports work to verify and explore this behavior. 4) Ricci Flow Near Kahler Geometries: For certain even-dimensional manifolds M, the set of Kahler geometries on M forms a subspace of the space of all Riemannian geometries on M. A Ricci flow solution which begins at a Kahler geometry remains Kahler. Are there Ricci flow solutions which begin outside the set of Kahler geometries but asymptotically approach it? The PI and his collaborators are working to show that this is the case, for a certain class of geometries.5) Stability of Neckpinch Behavior in Geometric Heat Flows: Neckpinch behavior in Ricci flow and mean curvature flow is well-understood for rotationally symmetric geometries and embeddings. There is evidence, both numerical and analytical, that such behavior is stable. The PI and his collaborators propose further work to verify this stability.