Geometric Boundary Value Problems in General Relativity

University Of Connecticut, Storrs CT

Investigators

Abstract

Many natural phenomena, such as the movement of liquids, the bending of solids, or the spread of temperature, can be described by partial differential equations (PDEs). For instance, electronic devices, including cell phones or computers, generate heat during normal operation, and it is necessary to conduct this heat away to prevent overheating. The Laplace equation, with specified boundary conditions like surface temperature, is used to analyze the steady-state of temperature distribution, ensuring efficient heat conduction and preventing overheating. Intriguingly, the study of our universe’s structure, governed by Einstein's general relativity, also gives rise to geometric PDEs similar to the Laplace equation. This research project aims to investigate those PDEs that arise from quantifying the mass or energy within bounded regions of the universe, such as glacial systems or binary black holes. The goal is to advance our understanding about the universe's structure by revealing hidden connections between the geometric boundary value problems and the known properties of the Laplace equation. The project will also involve mentoring students and conducting educational activities to enhance STEM awareness among a broader audience. The research project will address longstanding conjectures related to Bartnik’s quasi-local mass in general relativity and the existence of Einstein manifolds with prescribed boundary data. In 1989, Bartnik proposed a notion of quasi-local mass by minimizing the asymptotically defined masses among admissible extensions and raised several conjectures. Those conjectures have led to surprising connections with the positive mass theorem, the Penrose inequality, and scalar curvature. A novel approach has been applied to advance the Static Extension Conjecture and is expected to deepen our understanding of scalar curvature in the context of boundary geometry and the static vacuum manifolds. The recent significant progress toward the Stationary Conjecture and the pp-wave counter-examples in higher dimensions is anticipated to illuminate various mass rigidity problems, such as the hyperbolic positive mass theorem and general Penrose inequality. In addition, this research will resolve other geometric problems, particularly concerning the existence of Einstein manifolds with prescribed conformal boundary metrics and mean curvature. 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.