The primary topics of investigation for this proposal include the following important conjectures in Mathematical Relativity: the Static/Stationary Metric Extension Conjecture, the full Penrose Inequality and other geometric inequalities, as well as the Hoop Conjecture concerning black hole formation. The first of these arises from a particularly promising definition of quasilocal mass due to Bartnik, which seeks to localize the total (or ADM) mass. Although this definition satisfies most desired properties, and thus has the potential to be very useful, its abstract and nonconstructive nature yield severe limitations on understanding and applicability. In order to rectify this problem the Static/Stationary Metric Extension Conjecture asserts that the quasilocal mass may be calculated as the ADM mass of a solution to the Static/Stationary Vacuum Einstein equations which satisfies a certain geometric boundary condition. In recent joint work of the PI and M. Anderson, existence for this boundary value problem has been established under a reasonable assumptions; this development is a significant step towards the full conjecture. The Penrose Inequality relates the total mass of a spacetime to the area of its event horizons (boundary of black holes) via the inequality: total mass squared is greater than or equal to the total area of the event horizons divided by 16 pi. This may be viewed as a conjecture for an arbitrary spacelike slice of a spacetime, and in this setting an important special case (when the second fundamental form of the slice vanishes) has been confirmed independently by Huisken and Ilmanen (one black hole) and by Bray (finitely many black holes). In recent joint work with Bray, the PI has succeeded in reducing this problem to solving a canonical system of partial differential equations, and has confirmed existence in special cases. With Q. Han the PI has completely analyzed the blow-up and regularity properties for half of this system. It is a major goal of this project to complete this program by proving a general existence result, and thus establishing the Penrose Conjecture. The PI also plans to extend this reduction process in order to yield a systematic way of treating other related geometric inequalities.
Although there are numerous proposed definitions for quasilocal mass in General Relativity, most seem to suffer from one affliction or another. In the case of Bartnik's mass, the sole problem is the abstract nature of its definition. If the Static Extension Conjecture were verified, this lone difficulty should be resolved, thus opening the way for several applications. For example, it is generally believed that whenever enough mass is concentrated in too small of a region, gravitational collapse must ensue and result in a black hole - such a statement is often referred to as the Hoop Conjecture. While there are many ways to accurately describe the size of a region, until a proper notion of quasilocal mass is established, a rigorous and complete version of this conjecture will remain elusive. Another possible application of a properly defined notion of quasilocal mass, is to the long time existence problem for the Einstein Equations. As these equations form a hyperbolic system, it is tempting to search for a theory based on an energy method analogous to the classical theory of the scalar wave equation. A major obstacle to realizing such an approach, is the lack of an appropriate notion of energy for the gravitational field (a constituent of quasilocal mass). Moreover the boundary value problem associated with the Static/Stationary Extension Conjecture, is in fact an elliptic boundary value problem for the Ricci operator. Hence the techniques developed to study this conjecture should prove useful when investigating well-posed boundary value problems in several other settings, such as for instance, the Ricci flow on manifolds with boundary. As for the Penrose Inequality, it was originally put forth by Penrose to study what is perhaps the most important open question in General Relativity today, namely the Cosmic Censorship Conjecture (whether spacetime singularities are always enclosed by black holes), which is related to General Relativity's veracity as a physical theory. Heuristically, the Penrose Inequality is essentially a necessary condition for cosmic censorship to hold. Thus if the Penrose Inequality were to be confirmed it would add significantly to the general belief in the validity of cosmic censorship. Lastly, methods developed for the full Penrose Inequality are expected to provide a new powerful tool for reducing questions concerning general initial data sets to the case of time symmetry, and therefore will have numerous applications to a wide range of other problems in General Relativity.
