Illinois Journal of Mathematics

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

Theodoros Vlachos

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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

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

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

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Zentralblatt MATH identifier

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]


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.

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