Mean Field Equation of Liouville Type with Singular Data: Topological Degree - Chen, Chiun-Chuan; Lin, Chang-Shou (Communications on pure and applied mathematics, 2015)
We consider the following mean field equation: Δgv+ρ(h*ev∫Mh*ev−1)=4π∑j=1Nαj(δqj−1) on M, where M is a compact Riemann surface with volume 1, h* is a positive C1 function on M, and ρ and αj are constants satisfying αj > −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.