Differential Geometric Problems in Mathematical Relativity

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The remarkable detection, announced by LIGO in 2016, of gravitational waves generated by distant black holes has once again affirmed the extraordinary power of Einstein's General Theory of Relativity. General relativity is a geometric theory of gravity; in this theory the effects of gravity are due to the curvature of the universe. The gravitational field and other fields, black holes and related objects, may be described and analyzed using geometric methods. In very 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 (i.e., the large scale behavior of light rays and light cones). An initial data set in a spacetime consists of a smooth spacelike hypersurface, its induced metric and second fundamental form. The principal investigator (PI) will continue his investigations concerning asyptotically flat initial data sets with horizons. Such initial data sets model isolated gravitating systems with black holes. In this context the PI, with various collaborators, has established results on 'topological censorship' at the initial data level, for initial data sets in three and higher dimensions. These results establish, under natural physical assumptions, restrictions on the geometry and topology of the spatial domain of outer communication in black hole spacetimes. They rest in large measure on the rather recently developed theory of marginally outer trapped surfaces (MOTSs). These are objects of considerable interest. On the one hand they play an important role in the theory of black holes, and on the other, they may be viewed as natural spacetime analogues of minimal surfaces in Riemannian geometry. Although MOTS are not in general variational objects, they nevertheless turn out to have properties remarkably similar to minimal surfaces. The work of the PI, and related results, lead to many interesting open problems that the PI will investigate, including problems concerning the topology of MOTSs and the spatial domain of outer communication, as well as problems concerning MOTS existence and rigidity phenomena, both in the compact and noncompact case. Topological issues will benefit from a deeper understanding of the relationship between MOTSs and so-called immersed MOTSs. The PI will also work on some problems pertaining to `time-symmetric' (i.e., totally geodesic) initial data sets. The PI will implement a strategy for proving a positive mass theorem for asymptotically hyperbolic (AH) manifolds, in the spirit of the Schoen-Yau positive mass proof in the asymptotically Euclidean case. The research has the potential to close a gap in previous positive mass proofs in the AH case. A further aim in this direction would then be to try to use this method to obtain a proof of the Horowitz-Myers conjecture concerning the AdS soliton. The PI will also continue on-going projects involving photon spheres and spacetime rigidity problems.