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

• Published in: Communications on pure and applied mathematics Vol. 68; no. 6; pp. 887 - 947
• Main Authors: Chen, Chiun-Chuan, Lin, Chang-Shou
• Format: Journal Article
• Language: English
• Published: New York Blackwell Publishing Ltd 01.06.2015; John Wiley and Sons, Limited
• Subjects: Applied mathematics; Gravity; Topology
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.21532

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.