Metric Differential Geometry and Mathematical Gravity

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AbstractAward: DMS-0405906Principal Investigator: Gregory J. GallowayRecent developments in gravitational theory, both theoretical andempirical, have lead to an increased interest in the study ofasymptotically anti-de Sitter and asymptotically de Sitterspacetimes. In this project, techniques of Lorentzian andRiemannian geometry will be used to attack certain problemspertaining to such spacetimes, as well as some problemsconcerning new developments in the theory of black holes. One ofthe specific aims of this project is to continue development ofan approach to establishing positivity of mass for asymptoticallyhyperbolic Riemannian manifolds (with spherical conformalinfinity) which does not require spin assumption. This approach,which makes use of the brane action introduced by Witten and Yauin their work on the AdS/CFT correspondence, may also be usefulin settling some positive mass conjectures for asymptoticallyanti-de Sitter spacetimes with toroidal topology at infinity. ThePI also plans to continue investigations into uniqueness andrigidity phenomena, and the development of singularities inasymptotically de Sitter spacetimes. The classical definition ofa black hole, though mathematically elegant, is problematic inpractice (especially for numerical studies) in that it requiresknowing the full future evolution of spacetime all the way toinfinity. As an alternative, Ashtekar and Krishnan haveintroduced a new, fully nonlinear and dynamical, approach to thestudy of black holes, based on the quasi-local notion of adynamical horizon. Another aim of this project is to investigatesome open problems concerning the occurence, i.e., existence,uniqueness and position of dynamical horizons.Modern theories of gravity are geometrical in nature. Thegravitational field and other fields, black holes and relatedobjects, may be described and analyzed using geometric methods.In more general terms, this project is concerned with the studyof certain features of gravity of current scientific interestfrom this geometric point of view, utilizing the tools ofRiemannian geometry, a mathematical theory of space, andLorentzian geometry, a mathematical theory of spacetime. Thesetheories provide a method for studying the relationship amongthree fundamental aspects of the spacetime universe: curvature(i.e., the bending of space or spacetime), topology (i.e., theglobal shape and complexity of space or spacetime) and causalstructure (i.e., the large scale behavior of light rays andlight cones).