A Remark on the Use of Differential Forms for the Skyrme Problem

Abstract

We introduce a formulation of the Skyrme problem using differential forms. By means of this formulation, we prove first that the homothetic map between the standard three-sphere of radius R, S3r ⊂ R4, and S31 is the unique minimizer, modulo isometries, of the Skyrme energy in its homotopy class, for any R less than some critical value R0 ∈ (√3/2, √2]. We then establish a stability result for this Skyrme-form problem from which we can recover the result of M. Loss and N. S. Manton which states that this homothetic map is stable only up to R = √2.