## Illinois Journal of Mathematics

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

Theodoros Vlachos

#### 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
Revised: 9 October 2014
First available in Project Euclid: 7 July 2015

https://projecteuclid.org/euclid.ijm/1436275488

Digital Object Identifier
doi:10.1215/ijm/1436275488

Mathematical Reviews number (MathSciNet)
MR3367653

Zentralblatt MATH identifier
1322.53063

#### 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

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