On the Existence of Solutions to the Seiberg-Witten Vortex Equations with Exponential Decay on the Plane

Abstract

Clifford Taubes showed that the moduli space of the variational equations of the Yang-Mills-Higgs functional on the plane is non-empty, and its elements correspond to "vortices". Inspired by this result, in this paper, we completely characterize exponential decay solutions to the Seiberg-Witten vortex equations, where the equations are derived by applying a Hitchin-type dimensional reduction. We show that the existence of Seiberg-Witten ``vortices'' is not provided by the exponential decay solutions. This is in contrast to the Yang-Mill-Higgs vortex equations.

The introduction, Section 4, and Section 6 (old Section 5) are significantly revised. The previous main result in Section 4 was incorrect. This version updates the statement of the main theorem to include a uniqueness and nonexistence result regarding (non)singular sinh-Gordon equation. A new Section 5 has been added that highlights an interesting application to a constant mean curvature problem