Noncommutative Geometry Seminar

Date: September 28, 2022

Time: 2:00PM - 3:00PM

Location: BLOC 302

Speaker: Rudolf Zeidler, University of Munster

Title: Nonnegative scalar curvature on manifolds with at least two ends

Abstract: I will present an obstruction to positive scalar curvature (psc) on complete manifolds with at least two ends based on the existence of incompressible hypersurfaces that do not admit psc. This result mixes an analytic technique based on $\mu$-bubbles, an augmentation of the classical minimal hypersurface obstructions to psc, with a topological argument based on positive scalar curvature surgery. Due to the latter a surprising (but necessary!) spin condition enters our result even though our methods are not based on the Dirac operator. Concretely, let $M$ be an orientable connected $n$-dimensional manifold with $n\in\{6,7\}$ and $Y\subset M$ a two-sided closed connected incompressible hypersurface that does not admit a metric of psc. Suppose that the universal covers of $M$ and $Y$ are either both spin or both non-spin. Then $M$ does not admit a complete metric of psc. As a consequence, our result answers questions of Rosenberg-Stolz and Gromov up to dimension $7$. Joint work with Simone Cecchini and Daniel Räde.

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