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Mean Field Equation of Liouville Type with Singular Data: Topological Degree
Author(s) -
Chen ChiunChuan,
Lin ChangShou
Publication year - 2015
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21532
Subject(s) - mathematics , conic section , gravitational singularity , degree (music) , field (mathematics) , type (biology) , metric (unit) , curvature , mathematical analysis , pure mathematics , function (biology) , mathematical physics , geometry , physics , ecology , operations management , acoustics , economics , biology , evolutionary biology
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.