# Nonnegative scalar curvature on manifolds with at least two ends Source: [Nonnegative scalar curvature on manifolds with at least two ends]() Collection index: [llms.txt]() Data updated: 2025-10-25T16:35:48.631Z Authors: Simone Cecchini, Daniel R\\"ade, Rudolf Zeidler Published: 2022-05-24T16:17:13.000Z ## 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()