Geometry and Topology Seminar: Metric inequalities with positive scalar curvature

Abstract: We will discuss various situations where a certain perturbation of the Dirac operator on spin manifolds can be used to obtain distance estimates from lower scalar curvature bounds. A first situation consists in an area non-decreasing map from a Riemannian spin manifold with boundary X into the round sphere under the condition that the map is locally constant near the boundary and has nonzero degree. Here a positive lower bound of the scalar curvature is quantitatively related to the distance from the support of the differential of f and the boundary of X. A second situation consists in estimating the distance between the boundary components of Riemannian “bands” M × [−1,1] where M is a closed manifold that does not carry positive scalar curvature. Both situations originated from questions asked by Gromov. In the final part, I will compare the Dirac method with the minimal hypersurface method and show that if N is a closed manifold such that the cylinder N x R carries a complete metric of positive scalar curvature, then N also carries a metric of positive scalar curvature. This answers a question asked by Rosenberg and Stolz.

This talk is based on joint work with Daniel Räde and Rudolf Zeidler.

Host: Yanli Song