Some constructions of uniformly positive scalar curvature metrics on open manifolds

Mathematics > Differential Geometry [Submitted on 17 Jun 2026] Title:Some constructions of uniformly positive scalar curvature metrics on open manifolds View PDF HTML (experimental)Abstract:We obtain several constructions of uniformly positive scalar curvature complete Riemannian metrics on open manifolds. For dimension $n\geq3$, we show that if such a manifold admits a proper Morse function $f$ bounded below such that $f$ has no critical points of index $\geq n-2$, then it admits a uniformly positive scalar curvature metric. On the other hand if such a manifold admits a positive scalar curvature metric along with a compact exhaustion $\{U_i\}$ such that the boundary of each $U_i$ is minimal, then it also admits a uniformly positive scalar curvature metric. For dimension $4 \leq n\leq 7$, we show that if the manifold has product ends and a positive scalar curvature metric with $C$-quadratic decay at infinity for $C>4\pi^2$ with respect to some basepoint, then the existence of a mean convex hypersurface far enough from the basepoint implies the existence of a uniformly positive scalar curvature metric on the manifold. We study some applications of these results, including showing that if an open manifold of dimension $n\geq 3$ that admits no uniformly positive scalar curvature metric has a positive scalar curvature metric with mean convex exhaustion, then it admits a mean convex foliation of compact sets sufficiently close to the ends. On the other hand, if such a manifold has a mean concave exhaustion, then its ends admit a mean concave foliation. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.