Geometric quantities arising from bubbling analysis of mean field equations

2016/09/23 by Chang-Shou Lin, Chang‐Shou Lin, Lin, Chang-Shou +2
Computer Science · Mathematics · #14H70 #33E10 #35J08 #35J75 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Equations Stability Results #Nonlinear Partial Differential Equations #math.AP #msc:14H70 #msc:33E10 #msc:35J08 #msc:35J75

paper · pdf · doi:10.48550/arxiv.1609.07204

19 pages, no figures

arxiv created 2016/09/23 · openalex publication_date 2016/09/23 · arxiv updated 2016/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E = \Bbb C/Λ be a flat torus and G be its Green function with singularity at 0. Consider the multiple Green function G_n on Eⁿ: G_n(z₁,⋯,z_n) := ∑_{i < j} G(z_i - z_j) - n ∑_{i = 1} ⁿ G(z_i). A critical point a = (a₁, ⋯, a_n) of G_n is called trivial if \a₁, ⋯, a_n\ = \-a₁, ⋯, -a_n\. For such a point a, two geometric quantities D(a) and H(a) arising from bubbling analysis of mean field equations are introduced. D(a) is a global quantity measuring asymptotic expansion and H(a) is the Hessian of G_n at a. By way of geometry of Lamé curves developed in our previous paper (Cambridge J. Math 3, 2015), we derive precise formulas to relate these two quantities.

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