Stable Abelian Yang--Mills--Higgs Fields on the Round Three-Sphere

We prove that every stable weak critical point of the self-dual abelian Yang–Mills–Higgs energy on the round three-sphere is gauge equivalent to the vacuum, for every positive scale parameter. This establishes the three-dimensional statement proposed by Cheng. For the non-stable critical point, we give a quantitative upper bound for the lowest eigenvalue of the Hessian on the gauge quotient.