Mass Rigidity and Curvature Problems in Mathematical Relativity

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Einstein’s theory of gravity has been a strong driving force for the modern development in several branches of mathematics. Among its profound implications and wide applications, the theory of gravity successfully describes the shape of our universe and predicts celestial objects that were not known to exist, such as black holes. Over the past few decades, remarkable progress using advanced techniques in geometry and analysis has been made to resolve fundamental questions in general relativity, which has also led to the astonishing realization that some celestial objects are governed by the same mathematical principles as daily life objects, such as soap films. This project employs frontier developments in mathematics to investigate those interconnections and to further advance our understanding on the geometric structures of mathematical models of our universe. The project also incorporates mentoring and educational activities to promote geometry, analysis, and their interrelations with other STEM disciplines, to broader communities and to the society. Because of rapid advancement in geometric analysis in recent years, several longstanding questions in general relativity have been largely resolved. A prominent example is the resolution to the positive mass conjecture, including recent work on the spacetime positive mass theorem. At the same time, those resolutions in general relativity have motivated the development of new and unexpected techniques in geometry and analysis. The goal of this project is to interconnect general relativity with neighboring areas in geometry and analysis where some of innovative techniques can be further developed and applied. The scope of the project is to analyze curvature and geometric structure of initial data sets and their spacetime development, arising from mass minimization problems related to quasi-local mass, to investigate scalar curvature problems for compact manifolds and classification of static manifolds, and to characterize Einstein manifolds from the aspect of hyperbolic and conformal geometry. The project develops new ideas and techniques from differential geometry, analysis, partial differential equations, functional analysis, and calculus of variations to tackle fundamental questions in mathematical relativity and in the neighboring areas of geometry and analysis and is anticipated to have impact in other areas of mathematics and in theoretical physics.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.