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

A mix of human-written and AI-generated textHuman understanding: all partsmath.DG — Differential Geometrymath.AP — Analysis of PDEs

Contributed by Jiangcheng You ↗, Heng Zhang

SubmitterJiangcheng You

Version 1 / Oct 01, 2026 / CC BY 4.0

Abstract

We construct a smooth contractible open four-manifold that is not homeomorphic to R4\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.

Provenance statement

In a series of conversations concerning the above question, GPT-6 constructed a highly intricate counterexample and developed the associated proofs. Its contributions include the topological construction and the metric construction. The human researchers’ work consists of posing the research problem, simplifying the counterexample construction, completing the writing of this manuscript, and interpreting and verifying GPT-6’s arguments. The human authors take full responsibility for the content, accuracy, and integrity of this manuscript. The authors will not submit this manuscript for publication.

Tools used

OpenAI
ChatGPTVersion 6

References

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  2. O. Chodosh, D. M\'aximo, and A. Mukherjee, Complete Riemannian $4$-manifolds with uniformly positive scalar curvature , preprint, 2024, https://arxiv.org/abs/2407.05574 arXiv:2407.05574 [math.DG] . doi:\ https://doi.org/10.48550/arXiv.2407.05574 10.48550/arXiv.2407.05574 .arXivDOI
  3. S. Chang, S. Weinberger, and G. Yu, Taming $3$-manifolds using scalar curvature , Geom. Dedicata 148 (2010), 3--14. doi:\ https://doi.org/10.1007/s10711-009-9402-1 10.1007/s10711-009-9402-1 .DOI
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Version history

  1. v1Initial depositCurrentOct 01, 2026