On a Class of Conformal Metrics Arising in Work of Seiberg and Witten

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 xed compact Riemann surface. In consequence of these results, the Riemann sphere b C does not admit a Seiberg-Witten metric, but for all > 0 there is a conformal metric on b C of regularity C 2? which is Seiberg-Witten oo of a nite set. Theorem 1. Let ! be the area form of a conformal metric on a Riemann surface X. Then the following conditions are equivalent: (1) ! is locally of the form 1 ^ 2 ? 2 ^ 1 , where 1 and 2 are holomorphic (1,0)-forms. (2) For all p 2 X there exists a holomorphic coordinate z on a neighborhood of p together with a harmonic function h so that ! = i 2 h dz ^ dz. (3) On the domain of deenition of an arbitrary holomorphic coordinate z, ! takes the form i 2 e 2u dz ^ dz with u either a harmonic function or a smooth superharmonic function satisfying i@@ log ji@@uj + 4i@@u = 2 in the sense of distributions, a locally nite sum of delta masses. (4) There is a rank 2 vector bundle E ! X equipped with a holomorphically at Hermitian (1,1)-form-valued Lorentz metric g and a time-like holomorphic section : X ! E satisfying ! = g(;). (5) Either ! comes from a at metric, or else there are (a) a non-negative (1,1)-form on X inducing a conformal metric of curvature ?1 with isolated conical singularities having total angles of the form 2nn; n 2 and (b) a non-negative (1,1)-form on X inducing a at conformal metric with isolated conical singularities satisfying ! = 3=2 ?1=2 on fz 2 X : (z) 6 = 0g.