CAREER: Geometric Problems in General Relativity

University Of Connecticut, Storrs CT

Investigators

Abstract

Many fundamental results in mathematical general relativity concern the interplay between the globally conserved physical quantities and the geometric structure of our universe. This area of research in gravitational theory is highly active and requires new ideas from various fields of mathematics. The proposed research is important in understanding what geometric properties can be deduced from the Einstein equations and from the conserved quantities. The research projects may lead to the answers to some fundamental questions in general relativity, including rigidity of the spacetime positive mass theorem and geometric characterization of the center of mass. The research will employ techniques from differential geometry, partial differential equations, and geometric analysis. The project's educational activities will train a range of students into the field of geometric analysis and related areas. The main educational activities, including the Working Seminar, Geometry Day, and Summer Graduate Workshop, will prepare students for necessary backgrounds to start research in geometric analysis. Those activities will also attract students and faculty from other universities in the Northeast to participate and will encourage collaborations across universities. This research project comprises two research directions to better understand the globally conserved quantities in general relativity and their connections to the geometric structure. The first research direction concerns the moduli space of solutions to the Einstein constraint equations. A long term project is to fully understand how the conserved quantities, such as the ADM energy and linear momentum, vary on the moduli space. This project is inspired by studying rigidity of the positive mass conjecture and minimal mass extension of Bartnik's quasi-local mass. It is crucial to know how the conserved quantities vary under deformations. In addition, the PI will also study local deformation theorems for the constraint equations with dominant energy condition. The second project concerns the center of mass and angular momentum and the question of isoperimetry in asymptotically flat initial data sets. The PI has been investigating several different notions of center of mass and angular momentum, assuming parity conditions. It is desirable to continue to investigate the physical quantities for more general initial data sets. She also intends to study the stable constant mean curvature surfaces and isoperimetric surfaces that are naturally related to the geometric center of mass.