Differential Equations Seminar: Scalar curvature and codimension 2 collapse

This talk is about the structure of Riemannian 3-manifolds satisfying a lower bound on their scalar curvature. These manifolds are toy models for spatial geometry in general relativity. I will discuss a "drawstring" construction, which modifies a manifold near a given curve, reducing its length and incurring only negligible damage to a scalar curvature lower bound. This extends ideas of Basilio-Sormani and Lee-Naber-Neumayer. The drawstrings produce unexpected examples with relevance to a few areas, including the question "How flat is an isolated gravitational system with very little total mass?" This is based on joint work with Kai Xu.