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PreprintCC BY 4.0arXiv

A diameter gap for irreducible compact symmetric spaces of fixed rank

  • Zhiqi Chen
  • Shaoqiang Deng
  • Hui Zhang
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Abstract

We prove that, for every positive integer r, the positive normalized diameters of isometric quotients of simply connected irreducible compact Riemannian symmetric spaces of rank r have a uniform positive lower bound. The bound is independent of the dimension and of the acting group, which need not be connected. For the higher-dimensional Grassmannian families, we obtain an explicit bound for positive-dimensional acting groups by constructing nonconstant invariant functions of bounded trigonometric degree. The real and complex cases use moment maps. The quaternionic case follows from a quartic rigidity theorem proved by means of invariant four-forms and the classification of compact quaternionic symmetric spaces. We also give examples showing why the rank and the relative scales of product factors must be controlled.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026diameter,
  title = {{A diameter gap for irreducible compact symmetric spaces of fixed rank}},
  author = {Zhiqi Chen and Shaoqiang Deng and Hui Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.21535},
  eprint = {2609.21535},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21535},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A diameter gap for irreducible compact symmetric spaces of fixed rank
AU  - Zhiqi Chen
AU  - Shaoqiang Deng
AU  - Hui Zhang
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.21535
UR  - https://arxiv.org/abs/2609.21535
AB  - We prove that, for every positive integer r, the positive normalized diameters of isometric quotients of simply connected irreducible compact Riemannian symmetric spaces of rank r have a uniform positive lower bound. The bound is independent of the dimension and of the acting group, which need not be connected. For the higher-dimensional Grassmannian families, we obtain an explicit bound for positive-dimensional acting groups by constructing nonconstant invariant functions of bounded trigonometric degree. The real and complex cases use moment maps. The quaternionic case follows from a quartic rigidity theorem proved by means of invariant four-forms and the classification of compact quaternionic symmetric spaces. We also give examples showing why the rank and the relative scales of product factors must be controlled.
ER  -