Metric inequalities with positive scalar curvature

In recent years, distance inequalities on Riemannian manifolds under a positive scalar curvature bound have become an important aspect in the study of scalar curvature, in particular, due to many new ideas and conjectures raised by Gromov. In this talk, I will mostly focus on Riemannian bands, that is, Riemannian manifolds diffeomorphic to N✕J, where N is a closed manifold and J is an interval. In the first part, I will use a perturbation of the Dirac method to show that, if N is a closed, connected spin n-manifold with non-vanishing Rosenberg index and g is a metric on N✕[-1,1] whose scalar curvature is bounded from below by n(n-1), then the distance between the two boundary components of N✕[-1,1] is at most 2𝜋/n. This gives, in many geometrically relevant cases, a sharp answer to a question raised by Gromov. In the last part of the talk, I will use a perturbation of the minimal hypersurface technique to show the following result. Let N be a closed, oriented n-manifold, with n≤7 and n≠4. If the cylinder N✕(-∞,∞) carries a complete metric of positive scalar curvature, then N carries a metric of positive scalar curvature. This establishes, up to dimension 7, a conjecture due to Rosenberg and Stolz. The results using the Dirac method are joint work with Rudolf Zeidler. The results using the minimal hypersurface method are joint work with Daniel Räde and Rudolf Zeidler.