A Codimension Two Approach to the $\mathbb{S}^1$-Stability Conjecture

TL;DR

This paper proves Rosenberg's $ ext{S}^1$-stability conjecture for certain manifolds by introducing a new geometric bound related to the discrepancy between the circle's tangent and the normal vector field.

Contribution

It establishes the conjecture under a novel geometric condition measuring the discrepancy between tangent and normal vectors on the circle.

Findings

Proves the $ ext{S}^1$-stability conjecture under a geometric bound.

Identifies conditions where the conjecture holds despite known counterexamples.

Introduces a codimension two approach to the stability conjecture.

Abstract

J. Rosenberg's $\mathbb{S}^1$-stability conjecture states that a closed oriented manifold $X$ admits a positive scalar curvature metric iff $X\times \mathbb{S}^1$ admits a positive scalar curvature metric $h$. As pointed out by J. Rosenberg and others, there are known counterexamples in dimension four. We prove this conjecture whenever $h$ satisfies a geometric bound which measures the discrepancy between $\partial_\theta\in T\mathbb{S}^1$ and the normal vector field to $X\times \{P\}$, for a fixed $P\in \mathbb{S}^1.$