Oriented Zoll families of minimal hypersurfaces in ${\mathbb{C}\mathbb{P}}^n$ and ${\mathbb{H}\mathbb{P}}^n$

Let $M$ be either ${\mathbb{C}\mathbb{P}}^n$ or ${\mathbb{H}\mathbb{P}}^n$. We prove that the family $\bar{\mathcal{F}}=\{{\mathrm{\Sigma }}_{\sigma }{\}}_{\sigma \in M}$ of minimal geodesic spheres in $M$ characterizes the Fubini-Study metric $\bar{g}$, that is, $\bar{g}$ is the only metric that makes these real hypersurfaces minimal. This result is related to the recent theory of Zoll families of minimal hypersurfaces $(M,g,\mathcal{F}=\{{\mathrm{\Sigma }}_{\sigma }\})\in \mathcal{Z}$. It is conjectured that the only Zoll metric in ${\mathbb{C}\mathbb{P}}^n$ is $\bar{g}$. We present evidence that suggests a similar rigidity of $({\mathbb{C}\mathbb{P}}^n,\bar{g},\bar{\mathcal{F}})$ in this context of minimal hypersurfaces. Namely, we show that a direct adaptation of Guillemin’s Theorem for conformal deformations of Zoll metrics in ${\mathbb{S}}^2$ is not possible for $({\mathbb{C}\mathbb{P}}^n,\bar{g},\bar{\mathcal{F}})$, when $p=2n-1$ is a prime number.