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

中文翻译：

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至少有两端的流形上的非负标量曲率

让$\mathrm{中号}$是一个可定向的连接$n$维流形与$n\epsilon \{6,7\}$然后让$是\subset \mathrm{中号}$是一个两侧封闭连通的不可压缩超曲面，不允许正标量曲率度量（缩写为 psc）。此外，假设通用覆盖$\mathrm{中号}$和$是$要么都是自旋，要么都是非自旋。使用格罗莫夫的$\mu$-气泡，我们证明$\mathrm{中号}$不承认 psc 的完整度量。我们提供了一个例子，表明自旋/非自旋假设不能从该结果的陈述中删除。这回答了格罗莫夫针对一大类案例提出的直至维度 7 的问题。此外，我们证明了余维 2 的子流形的相关结果。作为特殊情况，我们推断，如果$是$不承认 psc 的度量并且$暗淡（是）\ne 4$， 然后$\mathrm{中号}:=是\times \mathbb{右}$不带有完整的 psc 指标并且$氮:=是\times {\mathbb{右}}^2$不携带统一 psc 的完整度量，前提是$暗淡（\mathrm{中号}）⩽7$和$暗淡（氮）⩽7$， 分别。这解决了 Rosenberg 和 Stolz 在可定向流形情况下的猜想（直到 7 维）。