2020/02/17 by S. Cecchini, Cecchini, Simone · 3 citations
Mathematics · Physics and Astronomy · #53C21 (Primary) 19K56 #58J99 (Secondary) #Advanced Differential Geometry Research #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Noncommutative and Quantum Gravity Theories
paper · pdf · doi:10.48550/arxiv.2002.07131
openalex publication_date 2020/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop index theory on compact Riemannian spin manifolds with boundary in\nthe case when the topological information is encoded by bundles which are\nsupported away from the boundary. As a first application, we establish a "long\nneck principle" for a compact Riemannian spin n-manifold with boundary X,\nstating that if textrmscal(X)\≥ n(n-1) and there is a nonzero degree map\ninto the sphere f colon X\→ Sn which is strictly area decreasing, then the\ndistance between the support of textrmd f and the boundary of X is at\nmost \π/n. This answers, in the spin setting and for strictly area\ndecreasing maps, a question recently asked by Gromov. As a second application,\nwe consider a Riemannian manifold X obtained by removing k pairwise\ndisjoint embedded n-balls from a closed spin n-manifold Y. We show that\nif textrmscal(X)>\σ>0 and Y satisfies a certain condition expressed\nin terms of higher index theory, then the radius of a geodesic collar\nneighborhood of \∂ X is at most \π \√((n-1)/(n\σ)). Finally,\nwe consider the case of a Riemannian n-manifold V diffeomorphic to N\×\n[-1,1], with N a closed spin manifold with nonvanishing Rosenberg index. In\nthis case, we show that if textrmscal(V)\≥\σ>0, then the distance\nbetween the boundary components of V is at most 2\π\n\√((n-1)/(n\σ)). This last constant is sharp by an argument due to\nGromov.\n