Journal of the Mathematical Society of Japan

A class of minimal submanifolds in spheres

Abstract

We introduce a class of minimal submanifolds $M^n$, $n\geq 3$, 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 minimal submanifolds $M^3$ in $\mathbb{S}^5$ with constant scalar curvature.

Article information

Source
J. Math. Soc. Japan, Volume 69, Number 3 (2017), 1197-1212.

Dates
First available in Project Euclid: 12 July 2017

https://projecteuclid.org/euclid.jmsj/1499846523

Digital Object Identifier
doi:10.2969/jmsj/06931197

Mathematical Reviews number (MathSciNet)
MR3685041

Zentralblatt MATH identifier
06786994

Citation

DAJCZER, Marcos; VLACHOS, Theodoros. A class of minimal submanifolds in spheres. J. Math. Soc. Japan 69 (2017), no. 3, 1197--1212. doi:10.2969/jmsj/06931197. https://projecteuclid.org/euclid.jmsj/1499846523

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