Award: DMS 1313724, Principal Investigator: Gregory J. Galloway
The principal investigator proposes to continue his work on two basic research directions: (i) the geometry and topology of initial data sets in general relativity, and (ii) spacetime rigidity results.
An initial data set in a spacetime M consists of a smooth spacelike hypersurface V, its induced metric h and second fundamental form K. A 'marginally outer trapped surface' (MOTS) in V is a closed two-sided hypersurface whose out-going light rays have vanishing divergence. The presence of a MOTS signals a gravitationally extreme situation: Generically one expects the development of singularities and the formation of a black hole. MOTSs arose in a more purely mathematical context in the work of Schoen and Yau concerning the existence of solutions to Jang's equation, in connection with their proof of positivity of mass. In a time-symmetric (K =0) initial data set a MOTS is simply a (2-sided) minimal surface. An important theme in minimal surface theory for many years has been the use of minimal surfaces to study the topology of Riemannian manifolds. In a similar vein, motivated by results on topological censorship (which are global in time results), the PI and collaborators, M. Eichmair and D. Pollack, have recently studied the relationship between the topology of 3-dimensional asymptotically flat initial data sets and the occurrence of MOTSs. In particular, using a consequence of geometrization and recent existence results for MOTSs, they have shown that non-trivial topology implies the existence of (immersed) MOTSs. This can be interpreted as a non-time symmetric version of results of Meeks- Simons-Yau. This work raises many interesting questions that the PI proposes to work on, particularly in the context of higher dimensional initial data manifolds with inner horizon (MOTS boundary) which satisfy the 'dominant energy condition', a physically natural curvature condition. The approach advocated makes use of the connection between solutions of Jang's equation and MOTSs. Various rigidity problems will also be addressed, in connection with the aforementioned work, the PI's previous work with R. Schoen on the topology of black holes and the PI's recent work with Carlos Vega concerning a spacetime rigidity problem originally posed by Yau, a concrete version of which is known as the Bartnik splitting conjecture.
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 (i.e., the large scale behavior of light rays and light cones).
