The existence of nonminimal solutions of the Yang-Mills-Higgs equations over \(\mathbb{R}^ 3\) with arbitrary positive coupling constant (Q1328203)

This paper is based on the work of \textit{C. H. Taubes} [Commun. Math. Phys. 86, 257-298 (1982; Zbl 0514.58016)], who developed a form of Lyusternik-Schnirelmann theory in order to overcome the analytic difficulties presented by non-compactness and established the existence of a critical point with monopole number (magnetic charge) \[ k = -{1\over 4\pi} \text{tr }\int_{\mathbb{R}^ 3} F\wedge D_ A \phi = 0 \] for the Yang-Mills-Higgs functional with coupling constant \(\lambda = 0\). In the above formula \(A\) is a connection on a principal SU(2) bundle, \(\phi\) a section on the vector bundle \(E = \text{SU}(2) \times \mathbb{R}^ 3\) called the Higgs field, \(D_ A\) is covariant differentiation and \(F\) is the curvature of the connection \(A\), \(F = dA + A \wedge A\). The aim of the paper is to prove that the result of Taubes can be generalized, i.e. to show that there is a non-globally-minimizing solution with \(k = 0\) and \(\lambda > 0\), and moreover, to substantiate that this solution is a local minimum. In section 2 of the paper the authors introduce the following definitions and notations: configurations \(c = ({\mathbf A}, \phi)\) configuration space, \(C = \{c = ({\mathbf A},\phi)| {\mathbf A}(c) < \infty\) and \(\lim_{| x| \to \infty} | \phi(x)| = 1\}\), gradient of \({\mathbf A}\) at \(c\): \(\nabla{\mathbf A}_ c(\psi) ={d\over ds} {\mathbf A}(c + s \psi) |_{s=0}\), where \(\psi = (\omega,\eta)\), \(\omega \in \text{SU}(2) \otimes \bigwedge_ p {\mathcal T}^*\). A critical point of \(A\) is a configuration \(c \in C\) if \(\nabla {\mathbf A}_ c (\psi) \equiv 0\) on a compactly supported connection. In section 3 it is established that there exists \(c_ 0 \in C\) such that \({\mathbf A}(c_ 0) = \inf_{k\neq 0} {\mathbf A} (c) > 0\), i.e. the infimum the energy functional over all configurations with non-zero monopole number exists and is positive, and moreover, there exists a configuration \(c_ 0\) that minimizes the action \(\mathbf A(0)\) over all classes of configurations with \(k \neq 0\). Subsequently (section 4) the configuration is used in order to construct a non-trivial loop of configurations that acts as a generator of a non-trivial homotopy class \(\pi_ 1(C)\) on which the min-max procedures are applied (section 5).