Hopf solitons on compact manifolds

Hopf solitons in the Skyrme-Faddeev system on $R^3$ 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 $E\ge Q$ 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 $S^3$, $T^3$ or $S^2 \times S^1$. In addition, we review the more elementary baby-Skyrme system, with M being $S^2$ or $T^2$.