  A topological lower bound on the Skyrme energy which depends explicity 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.
