Skip to content
DraconiticResearch
Search Sources API
203,948 studies31 sourcesUpdated 43 min ago
PreprintarXiv licencearXiv

A proof of the Simon conjecture and curvature rigidities

  • Weiran Ding
  • Jianquan Ge
  • Fagui Li
Open at sourcePDFDOICite

Abstract

In this paper, we prove all gaps in the Simon conjecture for connected closed minimal surfaces immersed in unit spheres. Our proof uses the line-bundle ladder of Lin, Wang, and Xu and a Dolbeault computation to obtain two-sided ordered spectral comparisons on the two-sphere under Gaussian curvature Kg≥1 and Kg≤1, with rigidity in the equality case. Applying these comparisons at consecutive Calabi scales forces the Gaussian curvature to be one of the two endpoint values. We also establish the corresponding rigidity theorem for closed surfaces with parallel mean curvature vector. Finally, we prove a rotationally symmetric higher-dimensional analogue under a Ricci lower bound and a sectional-curvature upper bound.

Related

  1. A proof of the Murty--Simon conjecture on diameter-$2$-critical graphs
    Sanjana Das, Jacob Fox
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  2. A proof of the Cartan-Hadamard conjecture for small volumes under a Ricci curvature lower bound
    Marcos Agnoletto, Márcio Fabiano Da Silva, Stefano Nardulli +1
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  3. Separation for Schrödinger operators. A counterexample to Simon's conjecture
    M. A. Perelmuter
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  4. Lenses make many points: Pure point spectrum disproves Simon's L2 conjecture
    Simon Becker, Mengxuan Yang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsMathematical physicsarXiv
  5. A proof of the Stanley-Stembridge conjecture
    Tatsuyuki Hikita
    Journal of the American Mathematical SocietyPeer-reviewedLicence not statedFull text51 citedCombinatorics & discrete mathematicsOpenAlex

Cite

BibTeXRIS
BibTeX
@misc{ding2026proof,
  title = {{A proof of the Simon conjecture and curvature rigidities}},
  author = {Weiran Ding and Jianquan Ge and Fagui Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20180},
  eprint = {2609.20180},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20180},
}
RIS
TY  - UNPB
TI  - A proof of the Simon conjecture and curvature rigidities
AU  - Weiran Ding
AU  - Jianquan Ge
AU  - Fagui Li
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.20180
UR  - https://arxiv.org/abs/2609.20180
AB  - In this paper, we prove all gaps in the Simon conjecture for connected closed minimal surfaces immersed in unit spheres. Our proof uses the line-bundle ladder of Lin, Wang, and Xu and a Dolbeault computation to obtain two-sided ordered spectral comparisons on the two-sphere under Gaussian curvature K_g≥1 and K_g≤1, with rigidity in the equality case. Applying these comparisons at consecutive Calabi scales forces the Gaussian curvature to be one of the two endpoint values. We also establish the corresponding rigidity theorem for closed surfaces with parallel mean curvature vector. Finally, we prove a rotationally symmetric higher-dimensional analogue under a Ricci lower bound and a sectional-curvature upper bound.
ER  -