A class of minimal submanifolds in spheres

TL;DR

This paper introduces a new class of minimal submanifolds in spheres, explores their deformation properties, and provides explicit examples, especially in dimensions 3 and 4, with connections to classical geometric problems.

Contribution

It defines a novel class of minimal submanifolds ruled by totally geodesic spheres, analyzes their deformation families, and constructs explicit examples related to the Chern-do Carmo-Kobayashi problem.

Findings

Existence of a one-parameter family of minimal isometric deformations for simply-connected examples.

Classification of compact examples in dimensions 3 and 4 as sphere bundles over minimal surfaces.

New examples with constant scalar curvature related to classical minimal submanifold problems.

Abstract

We introduce a class of minimal submanfolds $M^n$, $n\geq 3$, 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.