Collapsing in Differential Geometry and the Einstein Flow

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

A geometric flow is a controlled way to smoothly deform a geometric object, such as a curve or a surface, or a 3-dimensional slice of spacetime equipped with a metric that measures length and angles. This project is about the Einstein flow, which deforms such slices in a way that, when the slices are stacked up, the result is 4-dimensional solution of the vacuum Einstein equations, the fundamental equations that govern our universe. Deducing something about the future (or past) state of the universe from information about the present is of evident interest in cosmology, and pose a very challenging mathematical problem. In this project, the investigator will focus on the asymptotic (or long-term) behavior of spacetimes, in the "cosmological" setting where the slices are compact (i.e., they wrap back on themselves, like the surface of a balloon, no matter how large their size). In this context, even if the spacetime does not develop singularities (e.g., a black hole), the spatial slices can asymptotically collapse, meaning that their volumes become smaller than a naive rescaling argument would suggest. Recently, the investigator adapted techniques from Riemannian geometry (and in particular the study of collapsing solutions of the Ricci flow) to give new information about expanding vacuum spacetimes. Part of this project will be to build on these results and obtain more precise results about the future asymptotics of these spacetimes. Another component of this project will be the training of graduate students and postdoctoral scholars. Results of this project will also be disseminated to the public via journal publications, conferences, and posting to online mathematics archives. Collapsing in differential geometry is the phenomenon that a sequence of Riemannian manifolds can converge to a lower dimensional space in the Gromov-Hausdorff topology. The research in this proposal will extend collapsing methods, both within differential geometry and within geometric flows. In recent work, the investigator adapted collapsing techniques to give new information about the future asymptotics of expanding Einstein flows. One feature was the avoidance of any a priori symmetry assumptions; instead, continuous symmetries appeared in the collapsing limit. The proposed research will extend this in various ways. One direction is a more concrete understanding of the asymptotics of expanding vacuum solutions. A second direction is the asymptotics of shrinking Einstein flows, which are relevant for spacetime singularities. In addition, the investigator will work on problems in differential geometry and geometric flows such as collapsing with a lower bound on the curvature operator, and the behavior of scalar curvature lower bounds under metric convergence. 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.