Brian Allen
Abstract
Although scalar curvature is the simplest curvature invariant, our understanding of scalar curvature has not matured to the same level as Ricci or sectional curvature. Despite this fact, many rigidity phenomenon have been established which give some of the strongest insights into scalar curvature. Important examples include Geroch's conjecture, the positive mass theorem, and Llarull's theorem. In order to further understand scalar curvature we ask corresponding geometric stability questions, where the hypotheses of the rigidity phenomenon are relaxed, and one would like to show that Riemannian manifolds which satisfy the relaxed conditions are close to the rigid objects in some topology. In this note we will survey what is known for scalar curvature stability, discuss what the questions are in this area, and introduce important tools which have been useful so far.