Near BPS Skyrmions and restricted harmonic maps

Motivated by a class of near BPS Skyrme models introduced by Adam, Sánchez-Guillén and Wereszczyński, the following variant of the harmonic map problem is introduced: a map φ:(M,g)→(N,h)φ:(M,g)→(N,h) between Riemannian manifolds is restricted harmonic if it locally extremizes E2E2 on its View the MathML sourceSDiff(M) orbit, where View the MathML sourceSDiff(M) denotes the group of volume preserving diffeomorphisms of (M,g)(M,g), and E2E2 denotes the Dirichlet energy. It is conjectured that near BPS skyrmions tend to restricted harmonic maps in the BPS limit. It is shown that φφ is restricted harmonic if and only if φ∗hφ∗h has exact divergence, and a linear stability theory of restricted harmonic maps is developed, from which it follows that all weakly conformal maps are stable restricted harmonic. Examples of restricted harmonic maps in every degree class R3→SU(2)R3→SU(2) and R2→S2R2→S2 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 restricted harmonic, casting doubt on the phenomenological predictions of such studies. The problem of minimizing E2E2 for φ:Rk→Nφ:Rk→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 FF-criticality where FF is any functional on maps (M,g)→(N,h)(M,g)→(N,h) which is, in a precise sense, geometrically natural. The case where FF is a linear combination of E2E2 and E4E4, the usual Skyrme term, is studied in detail, and it is shown that inverse stereographic projection R3→S3≡SU(2)R3→S3≡SU(2) is stable restricted FF-critical for every such FF.