A Generalization of the Geroch Conjecture with Arbitrary Ends

TL;DR

This paper extends the Geroch conjecture by proving non-existence of positive scalar and intermediate curvature metrics on certain connected sums involving Schoen-Yau-Schick manifolds and arbitrary manifolds, using $$-bubbles.

Contribution

It introduces new non-existence results for metrics of positive scalar and intermediate curvature on connected sums, generalizing the Geroch conjecture to broader settings.

Findings

Connected sums with Schoen-Yau-Schick manifolds do not admit positive scalar curvature.

Certain connected sums do not admit metrics of positive intermediate curvature.

The work applies $$-bubbles to establish curvature obstructions.

Abstract

Using $\mu$-bubbles, we prove that for $3 \le n \le 7$, the connected sum of a Schoen-Yau-Schick $n$-manifold with an arbitrary manifold does not admit a complete metric of positive scalar curvature. When either $3 \le n \le 5$, $1 \le m \le n-1$ or $6 \le n \le 7$, $m \in \{1, n-2, n-1\}$, we also show the connected sum $(M^{n-m}\times \mathbb{T}^m) \# X^n$ where $X$ is an arbitrary manifold does not admit a metric of positive $m$-intermediate curvature. Here $m$-intermediate curvature is a new notion of curvature introduced by Brendle, Hirsch and Johne interpolating between Ricci and scalar curvature.