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

We consider the following mean field equation:Δ g v + ρ (h * e v∫ Mh *e v− 1 ) = 4 π ∑ j = 1 Nα j( δ q j− 1 )   on   M , 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. © 2015 Wiley Periodicals, Inc.