Oriented Zoll families of minimal hypersurfaces in $\mathbb{CP}^n$ and $\mathbb{HP}^n$

Abstract

Let $M$ be either $\mathbb{CP}^n$ or $\mathbb{HP}^n$. We prove that the family $\overline{\mathcal{F}}=\{Σ_σ\}_{σ\in M}$ of minimal geodesic spheres in $M$ characterizes the Fubini-Study metric $\overline{g}$, that is, $\overline{g}$ is the only metric that makes these real hypersurfaces minimal. This result is related to the recent theory of Zoll families of minimal hypersurfaces $(M,g,\mathcal{F}=\{Σ_σ\})\in\mathcal{Z}$. It is conjectured that the only Zoll metric in $\mathbb{CP}^n$ is $\overline{g}$. We present evidence that suggests a similar rigidity of $(\mathbb{CP}^n,\overline{g},\overline{\mathcal{F}})$ 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 $(\mathbb{CP}^n,\overline{g},\overline{\mathcal{F}})$, when $p=2n-1$ is a prime number.