Nonnegative scalar curvature on manifolds with at least two ends

Abstract

Let be an orientable connected -dimensional manifold with and let be a two-sided closed connected incompressible hypersurface that does not admit a metric of positive scalar curvature (abbreviated by psc). Moreover, suppose that the universal covers of and are either both spin or both nonspin. Using Gromov's -bubbles, we show that does not admit a complete metric of psc. We provide an example showing that the spin/nonspin hypothesis cannot be dropped from the statement of this result. This answers, up to dimension 7, a question by Gromov for a large class of cases. Furthermore, we prove a related result for submanifolds of codimension 2. We deduce as special cases that, if does not admit a metric of psc and , then does not carry a complete metric of psc and does not carry a complete metric of uniformly psc, provided that and , respectively. This solves, up to dimension 7, a conjecture due to Rosenberg and Stolz in the case of orientable manifolds.