HEXAGONbeta

A proof of the second pinching theorem in dimension four

A mix of human-written and AI-generated textHuman understanding: all partsmath.DG — Differential Geometry

Contributed by JINXI ZHANG ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

Let M⁴ be a closed connected manifold minimally immersed in the unit sphere S⁵(1), and let h be its second fundamental form. We prove that constant S = |h|² > 4 implies S > 8. No constancy of tr(h³) is assumed. The proof combines integral curvature identities with a joint estimate for two contractions of the symmetrized Hessian. The remaining algebraic inequality is verified in Mathematica using exact rational arithmetic. The code and data are included in the supplementary material.

Tools used

Wolfram
Mathematica
OpenAI
Codex

References

  1. É. Cartan, Sur des familles remarquables d'hypersurfaces isoparamétriques dans les espaces sphériques , Math. Z. 45 (1939), 335--367, doi:10.1007/BF01580289 (https://doi.org/10.1007/BF01580289).DOI
  2. S.-S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length , in Functional Analysis and Related Fields , F. E. Browder (ed.), Springer, Berlin, Heidelberg, 1970, pp. 59--75, doi:10.1007/978-3-642-48272-4_2 (https://doi.org/10.1007/978-3-642-48272-4_2).DOI
  3. R. Chen, An improved second pinching theorem for minimal hypersurfaces with constant scalar curvature in spheres , preprint (2026), arXiv:2609.17586v1 (https://arxiv.org/abs/2609.17586v1).arXiv
  4. Q.-M. Cheng, F. Li and G. Wei, An improved result on Chern conjecture , preprint (2026), arXiv:2610.05276v1 (https://arxiv.org/abs/2610.05276v1).arXiv
  5. J. Ge, T. Liu, K. Luo and W. Yan, Rigidity of closed minimal hypersurfaces in $\Sph^5$ , preprint (2026), arXiv:2606.29246v1 (https://arxiv.org/abs/2606.29246v1).arXiv
  6. S. Guan, The second gap for minimal hypersurfaces in the unit sphere with constant cubic trace , preprint (2026), arXiv:2609.26852v1 (https://arxiv.org/abs/2609.26852v1).arXiv
  7. J. Ge, H. Tan, W. Yan and Y. Zhang, The second gap and rigidity in Chern's conjecture with constant cubic trace , preprint (2026), arXiv:2609.27661v1 (https://arxiv.org/abs/2609.27661v1).arXiv
  8. C. He, H. Xu and E. Zhao, Classification of closed minimal hypersurfaces with constant scalar curvature in $\Sph^5$ , preprint (2026), arXiv:2603.01181v1 (https://arxiv.org/abs/2603.01181v1).arXiv
  9. H. B. Lawson, Jr., Local rigidity theorems for minimal hypersurfaces , Ann. of Math. (2) 89 (1969), 187--197, doi:10.2307/1970816 (https://doi.org/10.2307/1970816).DOI
  10. L. Lei, H. Xu and Z. Xu, On Chern's conjecture for minimal hypersurfaces in spheres , preprint (2017), arXiv:1712.01175v1 (https://arxiv.org/abs/1712.01175v1).arXiv
Show all 16 referencesShow fewer references
  1. H. F. Münzner, Isoparametrische Hyperflächen in Sphären , Math. Ann. 251 (1980), 57--71, doi:10.1007/BF01420281 (https://doi.org/10.1007/BF01420281).DOI
  2. H. F. Münzner, Isoparametrische Hyperflächen in Sphären. II. Über die Zerlegung der Sphäre in Ballbündel , Math. Ann. 256 (1981), 215--232, doi:10.1007/BF01450799 (https://doi.org/10.1007/BF01450799).DOI
  3. C. K. Peng and C. L. Terng, Minimal hypersurfaces of spheres with constant scalar curvature , in Seminar on Minimal Submanifolds , E. Bombieri (ed.), Ann. of Math. Stud. 103 , Princeton University Press, Princeton, NJ, 1983, pp. 177--198.
  4. C. K. Peng and C. L. Terng, The scalar curvature of minimal hypersurfaces in spheres , Math. Ann. 266 (1983), 105--113, doi:10.1007/BF01458707 (https://doi.org/10.1007/BF01458707).DOI
  5. J. Simons, Minimal varieties in Riemannian manifolds , Ann. of Math. (2) 88 (1968), 62--105, doi:10.2307/1970556 (https://doi.org/10.2307/1970556).DOI
  6. Y. J. Suh and H. Y. Yang, The scalar curvature of minimal hypersurfaces in a unit sphere , Commun. Contemp. Math. 9 (2007), no. 2, 183--200, doi:10.1142/S021919970700237X (https://doi.org/10.1142/S021919970700237X).DOI

Version history

  1. v1Submitted by JINXI ZHANGInitial depositCurrentOct 19, 2026