Minimal submanifolds in spheres and complex-valued eigenfunctions

TL;DR

This paper introduces a novel method for constructing minimal submanifolds in spheres using complex eigenfunctions, providing new descriptions and proofs related to eigenfunctions on spheres.

Contribution

It presents a new approach for constructing minimal submanifolds in spheres and offers a novel proof characterizing $(bb,bc)$-eigenfunctions on spheres.

Findings

Descriptions of Clifford torus and Lawson surfaces via eigenfunctions

A new proof characterizing $(bb,bc)$-eigenfunctions

Establishment of a link between eigenfunctions and their squares

Abstract

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.