Scalar Curvature and Product Manifolds

2026/07/15 by Claude LeBrun
#math.DG

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Abstract

We display 4-dimensional counter-examples to a restricted form of Rosenberg's S¹-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 S¹ nevertheless do admit such metrics.

Citations

• Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature
• Minimal Varieties in Riemannian Manifolds

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• Complete Riemannian 4-manifolds with uniformly positive scalar curvature
• Positive scalar curvature and exotic structures on simply connected four manifolds
• Geography of 4-manifolds with positive scalar curvature
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