Mean Field Equation of Liouville Type with Singular Data: Topological Degree

We consider the following mean field equation: urn:x-wiley::media:cpa21532:cpa21532-math-0001 where M is a compact Riemann surface with volume 1, h * is a positive C 1 function on M , and ρ and α j are constants satisfying α j > −1. In this paper, we derive the topological‐degree‐counting formula for noncritical values of ρ . We also give several applications of this formula, including existence of the curvature + 1 metric with conic singularities, doubly periodic solutions of electroweak theory, and a special case of self‐gravitating strings.