Uniform stability of monopoles

Abstract.

Two proofs of uniform stability in energy norm of BPS monopoles in the Yang-Mills-Higgs equations on Minkowski space-time are presented. BPS monopoles are minimisers of the static Yang-Mills-Higgs functional. The space of monopoles is an infinite dimensional submanifold whose quotient by the group of gauge transformations is finite dimensional. The problem of proving the existence and regularity of monopoles which are closest to the solution at each time is also discussed. The first proof establishes the existence of monopoles which are closest in $L^2$; this is achieved using Schoen-Uhlenbeck regularity theory for harmonic maps, and works for solutions of fairly high regularity ($H^k_{loc}$ with $k\ge 4$). A uniform bound is then obtained for the distance function which gives the distance from the solution to the projected point in a certain norm, first defined by Taubes, which is related to the energy. The second proof involves the direct minimisation of the distance function with respect to which stability is proved. Although this function is minimised only subject to a closeness condition, this approach has the advantage of yielding a proof of uniform stability valid for solutions with only the regularity required for finite energy ($H^1_{loc}$). There are two principal sources of difficulty in the present problem: the fact that the continuous spectrum of the Hessian extends all the way to zero and the presence of the infinite dimensional group of gauge symmetries.