Oriented Zoll families of minimal hypersurfaces in ℂℙⁿ and ℍℙⁿ

@misc{guajardo2026oriented,
  title = {{Oriented Zoll families of minimal hypersurfaces in $\mathbbCP^n$ and $\mathbbHP^n$}},
  author = {Diego Guajardo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25379},
  eprint = {2610.25379},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25379},
}

TY  - UNPB
TI  - Oriented Zoll families of minimal hypersurfaces in ℂℙ^n and ℍℙ^n
AU  - Diego Guajardo
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25379
UR  - https://arxiv.org/abs/2610.25379
AB  - Let M be either ℂℙ^n or ℍℙ^n. We prove that the family ℱ̅={Σ_σ}_σ∈ M of minimal geodesic spheres in M characterizes the Fubini-Study metric g̅, that is, g̅ is the only metric that makes these real hypersurfaces minimal. This result is related to the recent theory of Zoll families of minimal hypersurfaces (M,g,ℱ={Σ_σ})∈𝒵. It is conjectured that the only Zoll metric in ℂℙ^n is g̅. We present evidence that suggests a similar rigidity of (ℂℙ^n,g̅,ℱ̅) in this context of minimal hypersurfaces. Namely, we show that a direct adaptation of Guillemin's Theorem for conformal deformations of Zoll metrics in 𝕊^2 is not possible for (ℂℙ^n,g̅,ℱ̅), when p=2n−1 is a prime number.
ER  -