Appears in collection : Not Only Scalar Curvature Seminar

I will explain a spinorial approach towards a comparison and rigidity principle involving scalar and mean curvature for certain warped products over intervals. This is motivated by recent scalar curvature comparison questions of Gromov, in particular distance estimates under lower scalar curvature bounds on Riemannian bands $M \times [-1,1]$ and Cecchini's long neck principle. I will also exhibit applications of these techniques in the context of the positive mass theorem with arbitrary ends. This talk is based on joint work with Simone Cecchini.