On Non-l 2 Solutions to the Seiberg–witten Equations

We show that a previous paper of Freund describing a solution to the Seiberg– Witten equations has a sign error rendering it a solution to a related but different set of equations. The non-L 2 nature of Freund's solution is discussed and clarified and we also construct a whole class of solutions to the Seiberg–Witten equations. § 1. Introduction With the introduction of the Seiberg–Witten equations [1] there come a wealth of results on four manifold theory and a new improved point of view on Donaldson theory with an Abelian gauge theory supplanting a non-Abelian one—cf. [2] for a review. An important vanishing theorem of [1], reminiscent of the Lichernowicz–Weitzenböck vanishing theorems, shows that there are no non-trivial solutions to the Seiberg–Witten equations on four manifolds with non-negative Riemannian scalar curvature. However one can have non-trivial solutions which are singular in some way—for example one could have a non-trivial solution which was not L 2 : in [3] Freund describes such a non-L 2 to the Seiberg–Witten equations on R 4. Unfortunately a sign discrepancy in [3] means that the expressions given there obey equations which differ from the Seiberg–Witten equations in a certain sign. These other equations also admit L 2 solutions as well as non-L 2 ones and so Freund's equations are fundamentally different from the Seiberg–Witten equations. In § 2 we describe the Seiberg–Witten equations in a fairly explicit manner so as to expose notational conventions and matters of signs. In section § 3 we give the details concerning Freund's work and then in section § 4 we give an L 2 solution of Freund's equations and a class of solutions to the Seiberg–Witten equations.