Minimal submanifolds in spheres and complex-valued eigenfunctions

A new approach for constructing minimal submanifolds of codimension 1 in the round spheres is proposed. In the case of ${\mathbb{S}}^3$ two immersions of the Clifford torus and all Lawson ${\tau }_{n,m}$ surfaces are described in terms of $(\lambda ,\mu )$-eigenfunctions. Also, a new proof of a theorem that describes $(\lambda ,\mu )$-eigenfunctions on sphere is obtained. This proof is based on a statement that a function $f$ is a $(\lambda ,\mu )$-eigenfunction if and only if $f$ and $f^2$ are eigenfunctions for the Laplace-Beltrami operator.