On a class of conformal metrics arising in the work of Seiberg and Witten

Abstract.

We examine a class of conformal metrics arising in the “N = 2 supersymmetric Yang-Mills theory” of Seiberg and Witten. We provide several alternate characterizations of this class of metrics and proceed to examine issues of existence and boundary behavior and to parameterize the collection of Seiberg-Witten metrics with isolated non-essential singularities on a fixed compact Riemann surface. In consequence of these results, the Riemann sphere $\hat{\mathbb{C}}$ does not admit a Seiberg-Witten metric, but for all $\epsilon>0$ there is a conformal metric on $\hat{\mathbb{C}}$ of regularity $C^{2-\epsilon}$ which is Seiberg-Witten off of a finite set.