Research in Lorentzian Geometry and Mathematical Relativity

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AbstractProposal: DMS-9803566Principal Investigator: Gregory GallowayResearch under this award will be conducted in the fields ofLorentzian geometry and mathematical relativity. One of the centralaims of the proposed research is to develop new techniques forstudying spacetime rigidity phenomena. The proposed research seeks toestablish maximum principles for null hypersurfaces and use these toobtain splitting theorems/rigidity results for spacetimes whichcontain null lines. Problems in the theory of black holes are also tobe investigated. These new techniques for studying null hypersurfaceswill make amenable the problem of rigorously establishing undernatural regularity assumptions Hawking's black hole area theorem. Theinvestigator will also continue research on topological censorship andits relationship to the topology of black holes, focusing on theconnection between asymptotic structure and black hole topology.In more general terms, this research project is concerned withachieving a better understanding of the structure of the spacetimeuniverse at both the cosmological scale and the scale of collapsingstars. The aim of spacetime geometry, as to be utilized and developedin this project, is to study the relationship between threefundamental aspects of the spacetime universe: curvature (which by theEinstein equations is generated by the presence of matter), topology(i.e., the global shape of spacetime) and causal structure (i.e., thebehavior of light cones). This research has applications to generalrelativity and gravitation theory.