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Near BPS skyrmions and restricted harmonic maps. (English) Zbl 1321.53077

Motivated by a class of near BPS Skyrme models introduced in [C. Adam, J. Sánchez-Guillén and A. Wereszczyński, “A Skyrme-type proposal for baryonic matter”, Phys. Lett. 691, No. 2, 105–110 (2010; doi:10.1016/j.physletb.2010.06.025)], the author defines and investigates a variant of the harmonic map problem. In this approach, a map \(\varphi:(M,g)\to(N,h)\) between Riemannian manifolds is called restricted harmonic if it locally extremizes \(E_2\) on its \(\mathrm{SDiff}(M)\) orbit, where \(\mathrm{SDiff}(M)\) 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 restricted harmonic maps in the BPS limit. It is shown that \(\varphi\) is restricted harmonic if and only if \(\varphi^*h\) 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 \(\mathbb{R}^3\to \mathrm{SU}(2)\) and \(\mathbb{R}^2\to S^2\) are constructed. It is proved 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 \(E_2\) for \(\mathbb{R}^k\to N\) over all linear volume preserving diffeomorphisms is solved explicitly, and a deformed axially symmetric family of Skyrme fields is constructed which are candidates for approximate near BPS skyrmions at low baryon number.

MSC:

53C43 Differential geometric aspects of harmonic maps
53C80 Applications of global differential geometry to the sciences
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