Nonnegative scalar curvature on manifolds with at least two ends

Abstract

Let $M$ be an orientable connected $n$-dimensional manifold with $n\in \{6,7\}$ and let $Y\subset M$ 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 $M$ and $Y$ are either both spin or both nonspin. Using Gromov's $\mu$-bubbles, we show that $M$ 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 $Y$ does not admit a metric of psc and $dim\left(Y\right)\ne 4$, then $M:=Y\times R$ does not carry a complete metric of psc and $N:=Y\times R^2$ does not carry a complete metric of uniformly psc, provided that $dim\left(M\right)⩽7$ and $dim\left(N\right)⩽7$, respectively. This solves, up to dimension 7, a conjecture due to Rosenberg and Stolz in the case of orientable manifolds.