The Intrinsic Flat Metric between Oriented Rectifiable Metric Spaces and Applications"
Christina Sormani (CUNY, Graduate Center)
In the past 25 years great advances were made in the study of Riemannian manifolds and metric measure spaces endowed with a notion of lower bound on Ricci curvature by employing the
Gromov-Hausdorff distance and various notions of metric measure convergence. However, in
many settings one does not have such strong curvature controls including, for example, questions concerning manifolds with only lower bounds on scalar curvature or upper bounds on volume and diameter. In joint work with Stefan Wenger building upon work of Ambrosio-Kirchheim, we introduced the intrinsic flat distance between compact Riemannian manifolds and, more generally, oriented rectifiable metric spaces of finite Hausdorff measure. Wenger proved a compactness theorem requiring only a uniform upper bound on diameter and volume, and in joint work, we proved convergence with respect to this distance agrees with GH convergence on manifolds with nonnegative Ricci curvature. The relationship with smooth convergence away from singularities was explored in joint work with Sajjad Lakzian who then applied this to understanding Ricci flow through a rotationally symmetric neck pinch singularity. In joint work with Dan Lee, this notion has been applied to show asymptotically flat manifolds of nonnegative scalar curvature are close to Euclidean space if they have sufficiently small ADM mass. In current work in progress with Philippe LeFloch, the relationship between weak H^1 convergence of Riemannian metrics with uniform bounds on ADM mass and intrinsic flat convergence is being explored under rotational symmetry. Most recently, Raquel Perales has begun studying the intrinsic flat limits of Riemannian manifolds with boundary that have uniform control on boundary data without any assumptions on rotational symmetry.