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 which 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 non-spin. 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/non-spin 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 two. We deduce as special cases that, if $Y$ does not admit a metric of psc and $\dim(Y) \neq 4$, then $M := Y\times\mathbb{R}$ does not carry a complete metric of psc and $N := Y \times \mathbb{R}^2$ does not carry a complete metric of uniformly psc provided that $\dim(M) \leq 7$ and $\dim(N) \leq 7$, respectively. This solves, up to dimension $7$, a conjecture due to Rosenberg and Stolz in the case of orientable manifolds.