A long neck principle for Riemannian spin manifolds with positive scalar curvature
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GAFA Geometric And Functional Analysis
A LONG NECK PRINCIPLE FOR RIEMANNIAN SPIN MANIFOLDS WITH POSITIVE SCALAR CURVATURE Simone Cecchini
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 → S n which is strictly 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 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)/(nσ). 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 2π (n − 1)/(nσ). This last constant is sharp by an argument due to Gromov.
1 Introduction and main results The study of manifolds with positive scalar curvature has been a central topic in differential geometry in recent decades. On closed spin manifolds, the most powerful known obstruction to the existence of such metrics is based on the index theory for the spin Dirac operator. Indeed, the Lichnerowicz formula [Lic63] implies that, on a closed spin manifold Y with positive scalar curvature, the spin Dirac operator is invertible and hence its index must vanish. When X is a compact Riemannian manifold with boundary of dimension at least three, it is well known by classical results of Kazdan and Warner [KW75a, KW75b, KW75c] that X always carries a metric of positive scalar curvature. In order to use topological information to study metrics of positive scalar curvature on X, we need extra geometric conditions. When X is equipped with a Riemannian metric with a product structure near the boundary, it is well known [APS75a, APS75b, APS76] that the Dirac operator with global boundary conditions is elliptic. This fact has been
S. CECCHINI
GAFA
extensively used in the past decades to study metrics of positive scalar curvature in the spin setting. The purpose of this paper is to systematically extend the spin Dirac operator technique to the case when the metric does not necessarily have a product structure near the boundary and the topological information is encoded by bundles supported away from the boundary. As an application, we prove some metric inequalities with scalar curvature on spin manifolds with boundary, following the point of view recently
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