Illinois Journal of Mathematics

Isometric deformations of minimal surfaces in $\mathbb{S}^{4}$

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 provide an elementary proof of the fact that the space of all isometric minimal immersions $f:M\to\mathbb{S}^{4}$ of a 2-dimensional Riemannian manifold $M$ into $\mathbb{S}^{4}$ with the same normal curvature is, up to congruence, either finite or a circle. Furthermore, we show that if $M$ is compact and the Euler number of the normal bundle of $f$ is nonzero, then there exist at most finitely many noncongruent isometric minimal immersions of $M$ into $\mathbb{S}^{4}$ with the same normal curvature.

Article information

Source
Illinois J. Math., Volume 58, Number 2 (2014), 369-380.

Dates
Received: 26 March 2014
Revised: 9 October 2014
First available in Project Euclid: 7 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1436275488

Digital Object Identifier
doi:10.1215/ijm/1436275488

Mathematical Reviews number (MathSciNet)
MR3367653

Zentralblatt MATH identifier
1322.53063

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Citation

Vlachos, Theodoros. Isometric deformations of minimal surfaces in $\mathbb{S}^{4}$. Illinois J. Math. 58 (2014), no. 2, 369--380. doi:10.1215/ijm/1436275488. https://projecteuclid.org/euclid.ijm/1436275488


Export citation

References

  • A. C. Asperti, Immersions of surfaces into 4-dimensional spaces with nonzero normal curvature, Ann. Mat. Pura Appl. (4) 125 (1980), 313–328.
  • J. L. Barbosa, On minimal immersions of $\Sf ^{2}$ into $\Sf^{2m}$, Trans. Amer. Math. Soc. 210 (1975), 75–106.
  • S. S. Chern, On the minimal immersions of the two-sphere in a space of constant curvature, Problems in analysis, Princeton University Press, Princeton, 1970, pp. 27–40.
  • H. Choi, W. Meeks and B. White, A rigidity theorem for properly embedded minimal surfaces in $\mathbb{R}^{3}$, J. Differential Geom. 32 (1990), 65–76.
  • M. Dajczer and D. Gromoll, Gauss parametrizations and rigidity aspects of submanifolds, J. Differential Geom. 22 (1985), 1–12.
  • M. Dajczer and D. Gromoll, Real Kähler submanifolds and uniqueness of the Gauss map, J. Differential Geom. 22 (1985), 13–28.
  • J. H. Eschenburg, I. V. Guadalupe and R. Tribuzy, The fundamental equations of minimal surfaces in $\mathbb{C}P^{2}$, Math. Ann. 270 (1985), 571–598.
  • J. H. Eschenburg and P. Quast, The spectral parameter of pluriharmonic maps, Bull. Lond. Math. Soc. 42 (2010), 229–236.
  • J. H. Eschenburg and R. Tribuzy, Constant mean curvature surfaces in 4-space forms, Rend. Semin. Mat. Univ. Padova 79 (1988), 185–202.
  • I. V. Guadalupe and L. Rodriguez, Normal curvature of surfaces in space forms, Pacific J. Math. 106 (1983), 95–103.
  • R. Lashof and S. Smale, On the immersion of manifolds in euclidean space, Ann. of Math. (2) 68 (1958), 562–583.
  • J. Ramanathan, Rigidity of minimal surfaces in $\Sf^3$, Manuscripta Math. 60 (1988), 417–422.
  • B. Smyth and G. Tinaglia, The number of constant mean curvature isometric immersions of a surface, Comment. Math. Helv. 88 (2013), 163–183.
  • R. Tribuzy and I. V. Guadalupe, Minimal immersions of surfaces into 4-dimensional space forms, Rend. Semin. Mat. Univ. Padova 73 (1985), 1–13.
  • T. Vlachos, Congruence of minimal surfaces and higher fundamental forms, Manuscripta Math. 110 (2003), 77–91.