Colloquium - Simone Cecchini

Date: February 4, 2022

Time: 3:30PM - 4:30PM

Location: BLOC 117

Speaker: Simone Cecchini, University of Gottingen, Germany

Description:
Title: A long neck principle for Riemannian spin manifolds with positive scalar curvature
Abstract: We present results in index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a ``long neck principle'' for a compact Riemannian spin n-manifold with boundary X, stating that if scal(X) ≥ n(n-1) and there is a nonzero degree map f into the n-sphere which is area decreasing, then the distance between the support of the differential of f and the boundary of X is at most π/n. This answers, in the spin setting, a question asked by Gromov. As a second application, we consider the case of a Riemannian n-manifold V diffeomorphic to Nx [-1,1], where N is the (n-1)-torus or more in general a closed spin manifold with a suitable nonvanishing topological invariant. In this case, we show that, if scal(V) ≥ n(n-1), then the distance between the boundary components of V is at most 2π/n. This last constant is sharp by an argument due to Gromov.