Existence of energy minimizers as stable knotted solitons in the Faddeev model

In this paper, we study the existence of knot-like solitons realized as the energy-minimizing configurations in the Faddeev quantum field theory model. Topologically, these solitons are characterized by an Hopf invariant, Q, which is an integral class in the homotopy group π₃(S²) = ℤ. We prove in the full space situation that there exists an infinite subset script S sign of ℤ such that for any m ∈ script S sign, the Faddeev energy, E, has a minimizer among the topological class Q = m. Besides, we show that there always exists a least-positive-energy Faddeev soliton of non-zero Hopf invariant. In the bounded domain situation, we show that the existence of an energy minimizer holds for script S sign = ℤ. As a by-product, we obtain an important technical result which says that E and Q satisfy the sublinear inequality E ≤ C C|Q|^3/4, where C > 0 is a universal constant. Such a fact explains why knotted (clustered soliton) configurations are preferred over widely separated unknotted (multisoliton) configurations when |Q| is sufficiently large.