Journal of the Mathematical Society of Japan

A class of minimal submanifolds in spheres

Marcos DAJCZER and Theodoros VLACHOS

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1499846523

Digital Object Identifier
doi:10.2969/jmsj/06931197

Mathematical Reviews number (MathSciNet)
MR3685041

Zentralblatt MATH identifier
06786994

Subjects
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Secondary: 53B25: Local submanifolds [See also 53C40] 53C40: Global submanifolds [See also 53B25]

Keywords
minimal submanifolds ruled submanifolds isometric minimal deformations

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


Export citation

References

  • A. Asperti, Generic minimal surfaces, Math. Z., 200 (1989), 181–186.
  • J. Barbosa, On minimal immersions of $\mathbb{S}^2$ into $\mathbb{S}^{2m}$, Trans. Amer. Math. Soc., 210 (1975), 75–106.
  • R. Bryant, Submanifolds and special structures on the octonians, J. Differential Geom., 17 (1982), 185–232.
  • E. Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geom., 1 (1967), 111–125.
  • S. S. Chern, M. do Carmo and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length. 1970 Functional Analysis and Related Fields, Springer, New York, 59–75.
  • M. Dajczer and D. Gromoll, Real Kaehler submanifolds and uniqueness of the Gauss map, J. Differential Geom., 22 (1985), 13–28.
  • M. Dajczer and Th. Vlachos, The associated family of an elliptic surface and applications to minimal submanifolds, Geom. Dedicata, 178 (2015), 259–275.
  • M. Dajczer and Th. Vlachos, A class of complete minimal submanifolds and their associated families of genuine deformations, to appear in Comm. Anal. Geom.
  • M. Dajczer and Th. Vlachos, Isometric deformations of isotropic surfaces, Arch. Math., 106 (2016), 189–200.
  • J. H. Eschenburg and Th. Vlachos, Pseudoholomorphic Curves in $\mathbb{S}^6$ and the Octonions, preprint.
  • H. Hashimoto, T. Taniguchi and S. Udagawa, Constructions of almost complex 2-tori of type (III) in the nearly Kaehler 6-sphere, Differential Geom. Appl., 21 (2004), 127–145.
  • R. Miyaoka, The family of isometric superconformal harmonic maps and the affine Toda equations, J. Reine Angew. Math., 481 (1996), 1–25.
  • F. Urbano, Second variation of compact minimal Legendrian submanifolds of the sphere, Michigan Math. J., 51 (2003), 437–447.
  • Th. Vlachos, Minimal surfaces, Hopf differentials and the Ricci condition, Manuscripta Math., 126 (2008), 201–230.
  • Th. Vlachos, Exceptional minimal surfaces in spheres, Manuscripta Math., 150 (2016), 73–98.