Einstein Manifolds and General Relativity

Einstein's theory of general relativity describes gravity through the geometry of space and time. It is one of the greatest achievements in modern science, transforming our understanding of the universe and enabling revolutionary technologies such as GPS navigation. Yet many fundamental questions remain. How can we measure the gravitational energy contained within a region of space? Can we determine how gravity behaves inside a region by observing its boundary? When do the equations that describe gravity have stable solutions? This project will develop new mathematical methods to address these questions. The resulting advances will strengthen the foundations of general relativity, develop broadly applicable tools in geometry and analysis, and provide research training for students and early-career researchers. This project investigates the existence, characterization, and stability of Einstein metrics through three closely related research directions. The first focuses on Lorentzian Einstein metrics that minimize the Arnowitt–Deser–Misner (ADM) mass from the perspective of initial data. It seeks to establish the existence and uniqueness of Bartnik mass minimizers through new compactness results and analytical methods. The second examines the variational structure of the ADM mass functional, with particular emphasis on proving that static vacuum metrics are strict local minimizers of the ADM mass. The third develops an existence theory for Riemannian Einstein metrics with prescribed boundary data for compact and asymptotically hyperbolic manifolds. Collectively, these research directions will develop new analytical and geometric methods for studying Einstein equations in both Lorentzian and Riemannian settings. They will lead to fundamental advances in ADM mass minimization, stability, and geometric boundary value problems, with broad applications in geometric analysis and mathematical relativity. 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.