Spheres with transitive families of CMC hypersurfaces

Abstract

The goal of this paper is to investigate the following question: what geometric restrictions does the existence of a transitive family of constant mean curvature hyperspheres impose on a Riemannian metric on 𝕊ⁿ? The notion of a transitive family can be made precise through the theory of Zoll families developed by Ambrozio, Marques, and Neves. Within this framework, we partially extend their local rigidity theorem for even conformal factors on 𝕊³ to every dimension n≥4, assuming each hypersphere in the minimal Zoll family has the area of a round equator. Moreover, we give a partial answer to the following higher-dimensional version of a question posed by Robert Bryant: must a metric on 𝕊ⁿ, conformal to the round metric and admitting a Zoll family of hyperspheres with a common nonzero constant mean curvature, have constant sectional curvature? Finally, we also investigate inverse problems in the spirit of Hangan, asking which metrics make a fixed family of geodesic spheres have a common prescribed mean curvature.

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BibTeX RIS

BibTeX

@misc{miranda2026spheres,
  title = {{Spheres with transitive families of CMC hypersurfaces}},
  author = {Ivan Miranda and Luciano L. Junior},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08423},
  eprint = {2611.08423},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08423},
}

RIS

TY  - UNPB
TI  - Spheres with transitive families of CMC hypersurfaces
AU  - Ivan Miranda
AU  - Luciano L. Junior
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.08423
UR  - https://arxiv.org/abs/2611.08423
AB  - The goal of this paper is to investigate the following question: what geometric restrictions does the existence of a transitive family of constant mean curvature hyperspheres impose on a Riemannian metric on 𝕊^n? The notion of a transitive family can be made precise through the theory of Zoll families developed by Ambrozio, Marques, and Neves. Within this framework, we partially extend their local rigidity theorem for even conformal factors on 𝕊^3 to every dimension n≥4, assuming each hypersphere in the minimal Zoll family has the area of a round equator. Moreover, we give a partial answer to the following higher-dimensional version of a question posed by Robert Bryant: must a metric on 𝕊^n, conformal to the round metric and admitting a Zoll family of hyperspheres with a common nonzero constant mean curvature, have constant sectional curvature? Finally, we also investigate inverse problems in the spirit of Hangan, asking which metrics make a fixed family of geodesic spheres have a common prescribed mean curvature.
ER  -

Abstract

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