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math.DG — Differential Geometry

A contractible open four-manifold with a complete metric of uniformly positive scalar curvature

Contributed by Jiangcheng You, Heng Zhang

We construct a smooth contractible open four-manifold that is not homeomorphic to $\R^4$ and admits a complete Riemannian metric with scalar curvature at least one. This gives a negative answer to a question raised by Chang--Weinberger--Yu and Chodosh--M\'aximo--Mukherjee.

A mix of human-written and AI-generated textHuman understanding: all partsBaumslag–Solitar groupsGromov–Lawson surgerycontractible four-manifoldsgeometric topologyopen four-manifoldspositive scalar curvaturetopology at infinity

Oct 01, 2026PDF ↗View work →

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A contractible open four-manifold with a complete metric of uniformly positive scalar curvature

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