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 {{,mathrm{scal},}}(X)ge n(n-1) and there is a nonzero degree map into the sphere f:Xrightarrow S^n which is strictly area decreasing, then the distance between the support of text {d}f and the boundary of X is at most pi /n. This answers, in the spin setting and for strictly area decreasing maps, 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 {{,mathrm{scal},}}(X)>sigma >0 and Y satisfies a certain condition expressed in terms of higher index theory, then the radius of a geodesic collar neighborhood of partial X is at most pi sqrt{(n-1)/(nsigma )}. Finally, we consider the case of a Riemannian n-manifold V diffeomorphic to Ntimes [-1,1], with N a closed spin manifold with nonvanishing Rosenebrg index. In this case, we show that if {{,mathrm{scal},}}(V)ge sigma >0, then the distance between the boundary components of V is at most 2pi sqrt{(n-1)/(nsigma )}. This last constant is sharp by an argument due to Gromov.