A class of minimal submanifolds in spheres

Abstract

We introduce a class of minimal submanfolds $M^n$, $n\ge 3$, in spheres ${\mathbb{S}}^{n+2}$ that are ruled by totally geodesic spheres of dimension $n-2$. If simply-connected, such a submanifold admits a one-parameter associated family of equally ruled minimal isometric deformations that are genuine. As for compact examples, there are plenty of them but only for dimensions $n=3$ and $n=4$. In the first case, we have that $M^3$ must be a ${\mathbb{S}}^1$-bundle over a minimal torus $T^2$ in ${\mathbb{S}}^5$ and in the second case $M^4$ has to be a ${\mathbb{S}}^2$-bundle over a minimal sphere ${\mathbb{S}}^2$ in ${\mathbb{S}}^6$. In addition, we provide new examples in relation to the well-known Chern-do Carmo-Kobayashi problem since taking the torus $T^2$ to be flat yields a minimal submanifolds $M^3$ in ${\mathbb{S}}^5$ with constant scalar curvature.