Affiliation:
1. Texas A&M University Texas USA
2. Mathematisches Institut University of Göttingen Augsburg Germany
3. Institut für Mathematik Augsburg University Augsburg Germany
4. Mathematisches Institut University of Münster Münster Germany
Abstract
AbstractLet 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.
Funder
Studienstiftung des Deutschen Volkes
Deutsche Forschungsgemeinschaft
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