Hopf solitons on compact manifolds

Abstract

Hopf solitons in the Skyrme-Faddeev system on R3 typically have a complicated structure, in particular when the Hopf number Q is large. By contrast, if we work on a compact 3-manifold M, and the energy functional consists only of the Skyrme term (the strong-coupling limit), then the picture simplifies. There is a topological lower bound E ≥ Q on the energy, and the local minima of E can look simple even for large Q. The aim here is to describe and investigate some of these solutions, when M is S3, T3, or S2 × S1. In addition, we review the more elementary baby-Skyrme system, with M being S2 or T2.