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

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.