Let Y n be a closed oriented Schoen-Yau-Schick, enlargeable or 4-dimensional aspherical manifold. In this note we prove generalized band width estimates for Y n × [−1, 1], and collar width estimates for Y n × S 1 , with a point removed, in the presence of positive lower scalar curvature bounds and for n ≤ 6. These results are inspired by recent work of Cecchini and Zeidler in the spin setting and include the mean curvature of the boundary into the equation. We use results and techniques of Chodosh and Li, Gromov and Zhu regarding so called µ-bubbles, which in short are hypersurfaces minimizing a warped volume functional.

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Let M be an orientable connected n-dimensional manifold with n ∈ {6, 7} and let Y ⊂ 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 µ-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 ) = 4, then M := Y × R does not carry a complete metric of psc and N := Y × 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. * Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Priority Programme "Geometry at Infinity" (SPP 2026, CE 393/1-1).† Funded by a Doctoral Scholarship of the 'Studienstiftung des deutschen Volkes'.

Let 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.

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