Rigidity of minimal surfaces in S 3 (Q1101978)

Let M 2 be a two-dimensional Riemannian manifold and x: M \(2\to S\) 3 be an isometric immersion of M 2 into the unit sphere S \(3=\{x\in R\) 4; \(| x| =1\}\). The following theorem is proved: Assume that M 2 is compact and let x: M \(2\to S\) 3 be an isometric minimal immersion. Then there exist at most finitely many noncongruent minimal immersions with the same induced metric as x. The proof uses the family of minimal immersions associated to x, the fact that if the Gaussian curvature K of x satisfies \(K\not\equiv 1\), then the normal map of x is again minimal, and the fact that the coordinate functions of x are eigenfunctions of the Laplacian on M. An example is given to show that the above does not hold for minimal immersions of a torus T 2 into S 7.