Classification of harmonic mappings of constant energy density into spheres
Eigenmaps, i.e. harmonic maps of constant energy density, of a flat torus \({\mathbb{R}}^ m/\Lambda\), \(\Lambda\) is a lattice, into spheres can be parametrized by a compact convex body in a finite dimensional vector space in much the same way as in the work of Do Carmo and Wallach for minimal immersions of spheres and as extended by Li and the reviewer for irreducible homogeneous domains and eigenmaps, respectively. The main contribution here is the determination of the dimension of the parameter space in terms of the dual lattice of \(\Lambda\). This is then used to describe the eigenmaps and minimal immersions of the flat 2-torus into spheres very explicitly. In particular, unicity of Clifford tori in \(S^ 3\) due to Lawson and Sasaki is recovered. Further examples are given corresponding to \(m\geq 3\) and for the first two eigenvalues of \({\mathbb{R}}^ m/\Lambda\).
harmonic maps of constant energy density
scientific article; zbMATH DE number 4209127