HEXAGONbeta

Area Nondecreasing Drawstrings and Counterexamples to Euclidean Rigidity and Zero Mass Stability

Primarily human-written textHuman understanding: all partsmath.DG — Differential Geometry

Contributed by Jiangcheng You ↗, Heng Zhang

Version 1 / Oct 21, 2026 / CC BY 4.0

Abstract

Building on the drawstring construction of Kazaras--Xu, we develop a localized refinement that contracts prescribed curves while preserving positive scalar curvature and without decreasing two-dimensional area. As two applications, we disprove part~(b) of Gromov's Euclidean C0C^0 rigidity conjecture and Sormani's zero mass stability conjecture.

Provenance statement

ChatGPT 6 was employed to discuss potential ideas. Kimi K3 was used for grammatical refinement, spelling correction, and suggestions on the phrasing of section and lemma headings. Neither ChatGPT 6 nor Kimi K3 was used to produce any manuscript text, arguments, proofs, or computations. The authors take full responsibility for this work.

Tools used

OpenAI
ChatGPTVersion 6

References

  1. L. Ambrosio and B. Kirchheim, Currents in metric spaces , Acta Math. 185 (2000), 1--80.
  2. R. Arnowitt, S. Deser, and C. W. Misner, Coordinate invariance and energy expressions in general relativity , Phys. Rev. 122 (1961), 997--1006.
  3. R. H. Bamler, A Ricci flow proof of a result by Gromov on lower bounds for scalar curvature , Math. Res. Lett. 23 (2016), 325--337.
  4. R. Bartnik and J. Isenberg, The constraint equations , in P. T. Chruściel and H. Friedrich (eds.), The Einstein Equations and the Large Scale Behavior of Gravitational Fields , Birkhäuser, Basel, 2004, pp. 1--38.
  5. J. Basilio, J. Dodziuk, and C. Sormani, Sewing Riemannian manifolds with positive scalar curvature , J. Geom. Anal. 28 (2018), 3553--3602.
  6. H. L. Bray, Proof of the Riemannian Penrose inequality using the positive mass theorem , J. Differential Geom. 59 (2001), 177--267.
  7. S. Brendle, Constant mean curvature surfaces in warped product manifolds , Publ. Math. Inst. Hautes Études Sci. 117 (2013), 247--269.
  8. S. Brendle, Scalar curvature rigidity of convex polytopes , Invent. Math. 235 (2024), 669--708.
  9. P. Burkhardt-Guim, Pointwise lower scalar curvature bounds for C0C^0 metrics via regularizing Ricci flow , Geom. Funct. Anal. 29 (2019), 1703--1772.
  10. A. Carlotto, O. Chodosh, and M. Eichmair, Effective versions of the positive mass theorem , Invent. Math. 206 (2016), 975--1016.
Show all 29 referencesShow fewer references
  1. J. Chu, M.-C. Lee, and J. Wan, Rigidity of positive mass theorem with fast metric decay , arXiv:2607.17236v2 (https://arxiv.org/abs/2607.17236v2) (2026).arXiv
  2. C. Dong and A. Song, Stability of Euclidean 33-space for the positive mass theorem , Invent. Math. 239 (2025), 287--319.
  3. M. Gromov, Dirac and Plateau billiards in domains with corners , Cent. Eur. J. Math. 12 (2014), 1109--1156.
  4. M. Gromov, Four lectures on scalar curvature , arXiv:1908.10612v6 (https://arxiv.org/abs/1908.10612v6) (2021).arXiv
  5. D. Kazaras, M. Khuri, and D. A. Lee, Stability of the positive mass theorem under Ricci curvature lower bounds , Math. Res. Lett. 31 (2024), 747--794.
  6. D. Kazaras and K. Xu, Codimension 2 drawstrings with scalar curvature lower bounds , arXiv:2501.09149 (https://arxiv.org/abs/2501.09149) (2025).arXiv
  7. D. Kazaras and K. Xu, Drawstrings and flexibility in the Geroch conjecture , arXiv:2309.03756v3 (https://arxiv.org/abs/2309.03756v3) (2026).arXiv
  8. D. A. Lee and C. Sormani, Stability of the positive mass theorem for rotationally symmetric Riemannian manifolds , J. Reine Angew. Math. 686 (2014), 187--220.
  9. M.-C. Lee, Quantification of scalar curvature under C0C^0 convergence using smoothing , arXiv:2604.17759 (https://arxiv.org/abs/2604.17759) (2026).arXiv
  10. C. Li, A polyhedron comparison theorem for 33-manifolds with positive scalar curvature , Invent. Math. 219 (2020), 1--37.
  11. C. Li, Correction to: A polyhedron comparison theorem for 33-manifolds with positive scalar curvature , Invent. Math. 228 (2022), 535--538.
  12. L. Mazurowski and X. Yao, A positive mass theorem for continuous metrics , arXiv:2606.19123 (https://arxiv.org/abs/2606.19123) (2026).arXiv
  13. L. Mazurowski and X. Yao, Quantification of C0C^0 convergence in dimension three , arXiv:2604.14087 (https://arxiv.org/abs/2604.14087) (2026).arXiv
  14. L. Mazurowski and X. Yao, Scalar curvature under weak limits of manifolds , arXiv:2605.03136 (https://arxiv.org/abs/2605.03136) (2026).arXiv
  15. R. Schoen and S.-T. Yau, On the proof of the positive mass conjecture in general relativity , Comm. Math. Phys. 65 (1979), 45--76.
  16. C. Sormani, Geometric stability of the Schoen--Yau zero mass rigidity theorem , Surv. Differential Geom. 29 (2024), 309--353; arXiv:2604.17599v2 (https://arxiv.org/abs/2604.17599v2).arXiv
  17. C. Sormani and S. Wenger, The intrinsic flat distance between Riemannian manifolds and other integral current spaces , J. Differential Geom. 87 (2011), 117--199.
  18. E. Witten, A new proof of the positive energy theorem , Comm. Math. Phys. 80 (1981), 381--402.
  19. J. You and H. Zhang, Gromov's Euclidean endpoint C0C^0 rigidity for the positive mass theorem , arXiv:2606.15372v2 (https://arxiv.org/abs/2606.15372v2) (2026).arXiv

Version history

  1. v1Submitted by Jiangcheng YouInitial depositCurrentOct 21, 2026