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 $\dim (Y)\ne 4$, 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 $\dim (M)\le 7$ and $\dim (N)\le 7$, respectively. This solves, up to dimension $7$, a conjecture due to Rosenberg and Stolz in the case of orientable manifolds.

arXiv comment19 pages; v2: minor improvements. To appear in J. Topol

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Cited by in corpus (1)

• Rigidity and Non-Rigidity of ${\mathbb{H}}^n/{\mathbb{Z}}^{n-2}$ with Scalar Curvature Bounded from Below