Simone Cecchini, University of Göttingen

Title: A long neck principle for Riemannian spin manifolds with positive scalar curvature

Abstract: We develop 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
into the sphere f: X → Sⁿ which is area decreasing, then the distance between the support of df and the
boundary of X is at most π/n. This answers, in the spin setting, a question recently asked by
Gromov. As a second application, we consider a Riemannian manifold X obtained by removing
k pairwise disjoint embedded n-balls from a closed spin n-manifold Y. We show that if Scal(X)>σ>0 and Y
satisfies a certain condition expressed in terms of higher index theory, then the radius of a
geodesic collar neighborhood of ∂X is at most π[(n-1)(nσ)]^1/2. Finally, we consider the case of a Riemannian
n-manifold V diffeomorphic to Nx[-1,1], with N a closed spin manifold with nonvanishing
Rosenberg index. In this case, we show that if Scal(X)≥σ>0, then the distance between the boundary
components of V is at most π[(n-1)(nσ)]^1/2. This last constant is sharp by an argument due to Gromov.