Columbia Geometry & Analysis Seminar

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 NxR 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. Some of 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.