Topological energy bounds for the Skyrme and Faddeev models with massive pions

A topological lower bound on the Skyrme energy which depends explicitly on the pion mass is derived. This bound coincides with the previously best known bound when the pion mass vanishes, and improves on it whenever the pion mass is non-zero. The new bound can in particular circumstances be saturated. New energy bounds are also derived for the Skyrme model on a compact manifold, for the Faddeev–Skyrme model with a potential term, and for the Aratyn–Ferreira–Zimerman and Nicole models.