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

• Published in: Geometric and functional analysis Vol. 30; no. 5; pp. 1183 - 1223
• Main Author: Cecchini, Simone
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.10.2020
• Subjects: Analysis; Mathematics; Mathematics and Statistics
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-020-00545-1

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 → S n which is strictly area decreasing, then the distance between the support of d f and the boundary of X is at most π / 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 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 ) / ( n σ ) . Finally, we consider the case of a Riemannian n -manifold V diffeomorphic to N × [ - 1 , 1 ] , with N a closed spin manifold with nonvanishing Rosenebrg index. In this case, we show that if scal ( V ) ≥ σ > 0 , then the distance between the boundary components of V is at most 2 π ( n - 1 ) / ( n σ ) . This last constant is sharp by an argument due to Gromov.