On the Existence of Critical Points to the Seiberg-Witten functional

Let X be a closed smooth 4-manifold. In the Theory of the SeibergWitten Equations, the configuration space is Cα = Aα × Γ (S + α ), where Aα is a space of u1-connections defined on a complex line bundle over X and Γ (S α ) is the space of sections of the positive complex spinor bundle over X. The original SWα-equations are 1 -order PDE fitting into a variational principle SWα : Aα × Sα → R, which is invariant by the group action of (Gauge Group) Gα = Map(X,U1) and satisfies the Palais-Smale Condition, up to gauge equivalence. The Euler-Lagrange equations of the functional SWα are 2 -order PDE and the solutions of the original SWαequations are stable critical points. Our aim is to prove the existence of solutions to the Euler-Lagrange equations of the functional SWα by the method of the Minimax Principle.