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

Simone Cecchini  (Georg-August-Universitat Gottingen)

16:00-17:00, April 8, 2022   Zoom Meeting ID: 811 2195 8276 (Passcode: 115089)

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 N x [-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.

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