LEAP-MPS: Two Conjectures in Mathematical Relativity

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This project will use geometric methods to address two important conjectures in the mathematics of general relativity: the Penrose inequality conjecture, and the Horowitz-Myers conjecture. General relativity is a geometric theory of gravitation in which gravity can be explained by the curvature of spacetime. Perhaps the most fascinating prediction of general relativity is the existence of black holes – extremely dense and massive regions of spacetime where gravity is extremely powerful. The recent gravitational wave detection of black hole mergers by LIGO, and the `picture' of the M87 supermassive black hole by the EHT collaboration, confirm this prediction. In one part of the project, the PI will consider the Penrose inequality conjecture with angular momentum, which aims to clarify the relationship between the total energy, horizon area, and angular momentum of a rotating black hole. The PI will also work on the Horowitz-Myers conjecture, which would build on the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, a breakthrough in modern physics relating quantum field theories and supergravity theory. The project will provide opportunities for undergraduate students to work on research projects in the mathematics of general relativity, with an emphasis on the recruitment of students from underrepresented groups. Outreach activities hosted at Clark University will encourage local high school students to explore careers in STEM.In recent joint work with M. Khuri, H. Kunduri, and S.-T. Yau, the PI has studied singular harmonic maps to negatively curved manifolds and inverse curvature flows on Riemannian manifolds to prove several geometric inequalities involving (total or quasi-local) energy, angular momentum, horizon area, and charges of rotating black holes in general relativity. These results provide the foundation to address the Penrose inequality conjecture with angular momentum. On the other hand, in collaboration with P.-K. Hung and M. Khuri, the PI proved a positive energy theorem for asymptotically hyperboloidal initial data sets with toroidal infinity and smooth non-empty boundary using the new level set technique of spacetime harmonic functions. The PI will modify this technique to deal with complete asymptotically hyperbolic manifolds with toroidal infinity, which is the setting of the Horowitz-Myers conjecture.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.