Mass, Geometric Inequalities, and Partial Differential Equations in General Relativity

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The goal of this project is to study several important related conjectures in general relativity. This geometric theory of gravity is fundamental to our understanding of the large-scale structure of the universe, and has many practical applications such as to the fine tuning of global positioning system (GPS) technology. The PI will seek to establish families of geometric inequalities relating mass, charge, angular momentum, and horizon area, which probe the grand weak cosmic censorship conjecture. This conjecture asserts that whenever singularities arise in spacetime (which is a generic phenomenon) they must always be shrouded inside a black hole event horizon; this is intimately tied to whether general relativity is a proper deterministic theory. Special black hole solutions with symmetry (referred to as stationary axisymmetric and electro-vacuum) play a large role in our understanding of the theory, and this project seeks to classify them in higher dimensions relevant to string theory. In particular, the PI aims to prove existence and uniqueness for such black holes with exotic (lens space) topologies in five spacetime dimensions. Furthermore, new criteria for gravitational collapse and black hole formation will be studied, namely those due to concentration of angular momentum and/or charge. The PI will also examine proposed definitions of quasi-local mass in order to determine whether they are mathematically and physically viable.Based on earlier work with Bray, the PI has recently completed a systematic approach to treating the full family of Penrose-type inequalities by reducing each to a canonical system of elliptic partial differential equations (PDEs). Thus, the entire range of these geometric inequalities is within reach. A natural by-product, associated with the study of a certain subclass of these inequalities, concerns a general procedure for constructing singular harmonic maps having 2-dimensional hyperbolic space target, which naturally arise from the stationary axisymmetric vacuum Einstein equations in four dimensions. Together with G. Weinstein, who initiated the study of such harmonic maps with prescribed singularities, we have begun development of the tools necessary to substantially generalize the 4-dimensional results to allow for exotic topologies in higher dimensions as well as a wide range of symmetric space targets.  Moreover, in joint work with M. Anderson, the PI has established what may be considered as the first step of Bartnik's minimal mass extension conjecture, and our methods indicate what should be a successful approach to the remaining parts. The trapped surface/hoop conjecture, dealing with the conditions under which black holes may form, is highly sought after but not well understood. However, the PI's work on Penrose-type inequalities suggests new black hole formation criteria as well as certain isoperimetric-type inequalities for relativistic bodies.