Near BPS Skyrmions and Restricted Harmonic Maps

Motivated by a class of near BPS Skyrme models introduced by Adam, S\'anchez-Guill\'en and Wereszczy\'nski, the following variant of the harmonic map problem is introduced: a map $\phi:(M,g)\rightarrow (N,h)$ between Riemannian manifolds is restricted harmonic (RH) if it locally extremizes $E_2$ on its $SDiff(M)$ orbit, where $SDiff(M)$ denotes the group of volume preserving diffeomorphisms of $(M,g)$, and $E_2$ denotes the Dirichlet energy. It is conjectured that near BPS skyrmions tend to RH maps in the BPS limit. It is shown that $\phi$ is RH if and only if $\phi^*h$ has exact divergence, and a linear stability theory of RH maps is developed, whence it follows that all weakly conformal maps, for example, are stable RH. Examples of RH maps in every degree class $R^3\to SU(2)$ and $R^2\to S^2$ are constructed. It is shown that the axially symmetric BPS skyrmions on which all previous analytic studies of near BPS Skyrme models have been based, are not RH, so each such field can be deformed along $SDiff(R^3)$ to yield BPS skyrmions with lower $E_2$, casting doubt on the predictions of such studies. The problem of minimizing $E_2$ for $\phi:R^k\to N$ over all linear volume preserving diffeomorphisms is solved explicitly, and a deformed axially symmetric family of Skyrme fields constructed which are candidates for approximate near BPS skyrmions at low baryon number. The notion of restricted harmonicity is generalized to restricted $F$-criticality where $F$ is any functional on maps $(M,g)\to (N,h)$ which is, in a precise sense, geometrically natural. The case where $F$ is a linear combination of $E_2$ and $E_4$, the usual Skyrme term, is studied in detail, and it is shown that inverse stereographic projection $R^3\to S^3\equiv SU(2)$ is stable restricted $F$-critical for every such $F$.