Principal Investigator: Gregory J. Galloway
This proposal is concerned with research projects at the interface of spacetime geometry and general relativity. In particular, properties of marginally trapped surfaces, which have played an important role in theory of black holes, will be investigated. Although physically well motivated, very few rigorous results about marginally trapped surfaces, and associated objects, such as dynamical horizons, had been known til recently. This situation has changed, and there are now rigorous results concerning existence, uniqueness, and other issues. Motivating this study in part is the realization of a deeper connection between marginally trapped surfaces in spacetime and minimal surfaces in Riemannian manifolds. The PI will conduct research on a number of topics pertaining to marginally trapped surfaces, such as the topology of black holes, rigidity and regularity of outermost marginally trapped surfaces, and aspects of dynamical horizons. String theory has increased interest in gravity in higher dimensions, and, in particular, there has been a great deal of recent research concerning black holes in higher dimensional spacetimes. Schoen and the PI recently obtained a generalization to higher dimensions of a classical result of Hawking concerning the topology of black holes. One part of the present proposal, involving joint work with Schoen, concerns an effort to obtain an important strengthening of this result, which would close a gap in the conclusion concerning allowed topologies. This circle of ideas should also be useful in obtaining lower entropy bounds for higher dimensional stationary black holes in the asymptotically AdS setting that extend known results in four dimensions. The PI will also conduct research on a continuing project with Andersson and Cai concerning the positivity of mass for asymptotically hyperbolic manifolds, and research pertaining to a problem of Penrose concerning the classical stability of string theory.
Modern theories of gravity are geometrical in nature. The gravitational field and other fields, black holes and related objects, may be described and analyzed using geometric methods. In more general terms, this project is concerned with the study of certain features of gravity of current scientific interest from this geometric point of view, utilizing the tools of Riemannian geometry, a mathematical theory of space, and Lorentzian geometry, a mathematical theory of spacetime. These theories provide a method for studying the relationship among three fundamental aspects of the spacetime universe: curvature (i.e., the bending of space or spacetime), topology (i.e., the global shape and complexity of space or spacetime) and causal structure structure (i.e., the large scale behavior of light rays and light cones).
