Black Holes, Geometric Inequalities, and Partial Differential Equations

The goal of this project is to study several important related conjectures in general relativity. This geometric theory of gravity proposed by Einstein is fundamental to our understanding of the large scale structure of the universe, and has many practical applications such as to the fine tuning of global positioning system (GPS) technology. The PI will seek to establish families of geometric inequalities relating mass, charge, angular momentum, and horizon area, which probe the grand weak cosmic censorship conjecture.

This conjecture asserts that whenever singularities arise in spacetime (which is a generic phenomenon) they must always be shrouded inside a black hole event horizon; this is intimately tied to whether general relativity is a proper deterministic theory. Special black hole solutions with symmetry (referred to as stationary axisymmetric and electro-vacuum) play a large role in our understanding of the theory, and this project seeks to classify them in higher dimensions relevant to string theory.

Furthermore, new criteria for gravitational collapse and black hole formation will be studied, and the PI will also examine proposed definitions of quasi-local mass in order to determine whether they are mathematically and physically viable. Lastly, fundamental questions concerning the shape (topology) of the cosmos will be addressed, including whether we live in a finite or infinite universe.

Recently the PI, in collaboration with Bray, Kazaras, and Stern has found a new and simple proof of the positive mass theorem, a result which has played a seminal role in mathematical relativity since its initial proof by Schoen, Yau, and Witten 40-years ago. This new approach, which has also been generalized to the spacetime and hyperbolic settings, yields an explicit lower bound for the mass in terms of quantities associated with (spacetime) harmonic functions, and suggests a strategy to establish the conjectured stability or almost rigidity for this theorem.

Based on an initial collaboration with Bray, the PI has completed a systematic approach to treating the full family of Penrose-type inequalities by reducing each to a canonical system of elliptic PDE, thus placing the entire range of these geometric inequalities within reach. Together with Yamada and Weinstein, who initiated the study of harmonic maps with prescribed singularities associated with the axisymmetric 4D Einstein equations in the 1990s, the PI has developed the tools necessary to substantially generalize the 4D results to allow for exotic topologies in higher dimensions as well as a wide range of symmetric space targets.

This work suggests that it is possible to obtain the full classification of stationary axisymmetric black holes within vacuum and supergravity. In joint work with Alaee and Yau, the PI has begun a study of the proposed Bekenstein bounds based on the Wang-Yau quasi-local mass; these inequalities concern the entropy/information contained within a relativistic body and have wide ranging implications from thermodynamics to computer science.

This initial study has provided the foundations to address the full Bekenstein conjecture, and is closely related via the approach to the trapped surface/hoop conjecture dealing with the conditions under which black holes may form. In addition, joint work with Anderson has established what may be considered as the first step of Bartnik's minimal mass extension conjecture, and the PI's methods indicate what should be a successful strategy for the remaining parts.

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.