Scalar Curvature and Product Manifolds

Claude LeBrun

Scalar Curvature and Product Manifolds

Differential Geometry 2026-07-15 v1

Authors: Claude LeBrun

Abstract

We display 4-dimensional counter-examples to a restricted form of Rosenberg's S^1-Stability Conjecture recently proposed by Jie Xu. Namely, we show that there are smooth compact 4-manifolds M with Euler characteristic zero that do not admit positive-scalar-curvature Riemannian metrics, but for which the corresponding Cartesian products M x S^1 nevertheless do admit such metrics.

Keywords

Scalar Curvature Product Manifolds Rosenberg's Conjecture Counter-Examples 4-Manifolds

Cite

@article{arxiv.2607.14244,
  title  = {Scalar Curvature and Product Manifolds},
  author = {Claude LeBrun},
  journal= {arXiv preprint arXiv:2607.14244},
  year   = {2026}
}

Comments

11 pages, LaTex2e