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arXiv · 2412.12479

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

Steven Rosenberg·Jie Xu
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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 $P\in \mathbb{S}^1.$

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Steven Rosenberg, Jie Xu. 2024-12-17. A Codimension Two Approach to the $\mathbb{S}^1$-Stability Conjecture. https://arxiv.org/abs/2412.12479

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