Nonnegative scalar curvature on manifolds with at least two ends

TL;DR

This paper proves that certain high-dimensional manifolds with specific hypersurfaces cannot admit complete metrics of positive scalar curvature, addressing a question by Gromov and a conjecture by Rosenberg and Stolz in dimensions up to 7.

Contribution

It establishes new obstructions to positive scalar curvature on manifolds with multiple ends, using Gromov's μ-bubbles, and confirms conjectures for a broad class of cases in low dimensions.

Findings

Manifolds with at least two ends and certain hypersurfaces do not admit complete psc metrics.

Provides counterexamples showing the necessity of spin/non-spin conditions.

Extends results to submanifolds of codimension two and product manifolds.

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.