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Rigidity and flexibility under positive isotropic curvature

  • Tsz-Kiu Aaron Chow
  • Yipeng Wang
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Abstract

For every n≥4 and L>0, we construct a smooth $4$-PIC metric on Sn with Urysohn $1$-width at least L and an embedded stable minimal disk of intrinsic inradius at least L. These examples disprove the proposed width and stable-disk radius bounds under a positive lower bound for isotropic curvature. On closed even-dimensional manifolds, we prove the sharp estimate λ1(2)≥(n−1)σ/2 under σ-PIC and show that equality forces roundness if a closed eigenform attaining the bound has rank at least four at some point.

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Cite

BibTeXRIS
BibTeX
@misc{chow2026rigidity,
  title = {{Rigidity and flexibility under positive isotropic curvature}},
  author = {Tsz-Kiu Aaron Chow and Yipeng Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11702},
  eprint = {2609.11702},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11702},
}
RIS
TY  - UNPB
TI  - Rigidity and flexibility under positive isotropic curvature
AU  - Tsz-Kiu Aaron Chow
AU  - Yipeng Wang
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11702
UR  - https://arxiv.org/abs/2609.11702
AB  - For every n≥4 and L>0, we construct a smooth $4$-PIC metric on S^n with Urysohn $1$-width at least L and an embedded stable minimal disk of intrinsic inradius at least L. These examples disprove the proposed width and stable-disk radius bounds under a positive lower bound for isotropic curvature. On closed even-dimensional manifolds, we prove the sharp estimate λ_1^(2)≥(n−1)σ/2 under σ-PIC and show that equality forces roundness if a closed eigenform attaining the bound has rank at least four at some point.
ER  -