[Paper Review] Geometric quantities arising from bubbling analysis of mean field equations

Chang‐Shou Lin, Chin-Lung Wang|arXiv (Cornell University)|Sep 23, 2016

Nonlinear Partial Differential Equations7 references3 citations

TL;DR

This paper establishes a precise geometric relationship between two key quantities—D(a), which controls the asymptotic behavior of bubbling solutions to mean field equations on flat tori, and H(a), the Hessian of the multiple Green function at trivial critical points. Using Lamé curve theory and complex analysis, it derives explicit formulas linking D(a) and H(a), enabling the classification and uniqueness of bubbling solutions via their blowup configurations.

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 $E^n$: $$G_{n}(z_1,\cdots,z_n) := \sum_{i < j} G(z_{i} - z_{j}) - n \sum_{i = 1} ^{n} G(z_{i}).$$ A critical point $a = (a_1, \cdots, a_n)$ of $G_n$ is called trivial if $\{a_1, \cdots, a_n\} = \{-a_1, \cdots, -a_n\}$. 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.

Motivation & Objective

• To understand the geometric role of D(a) and H(a) in the bubbling analysis of mean field equations on flat tori.
• To resolve the question of when trivial critical points of the multiple Green function G_n are degenerate, by analyzing H(a).
• To derive explicit, computable formulas relating D(a) and H(a) using the geometry of Lamé curves.
• To provide a framework for determining local uniqueness of bubbling solutions via non-vanishing D(a) and H(a).

Proposed method

• Introduces D(a) as a global geometric quantity measuring the asymptotic expansion of ρ_k − 8πn in bubbling sequences.
• Defines H(a) as the Hessian determinant of the multiple Green function G_n at trivial critical points a.
• Applies the theory of Lamé curves to parametrize the critical points of G_n on E^n.
• Uses complex analysis and Weierstrass elliptic functions to compute second derivatives of the Green function G.
• Derives explicit formulas for det D^2 G_n at special points (half-periods and Weierstrass points) via (log ϑ)_{zz} and ϑ-function identities.
• Establishes a proportionality between D(a) and H(a) through a constant c_p, linking the two quantities via the geometry of the torus and its invariants.

Experimental results

Research questions

• RQ1 How are the geometric quantities D(a) and H(a) related in the context of bubbling solutions to mean field equations on flat tori?
• RQ2 What is the precise relationship between the asymptotic control parameter D(a) and the Hessian H(a) at trivial critical points of the multiple Green function?
• RQ3 Can the Hessian H(a) be computed explicitly at special points such as half-periods and Weierstrass points on the torus?
• RQ4 Under what conditions does the Hessian H(a) vanish, and how does this affect the uniqueness of bubbling solutions?
• RQ5 How does the geometry of Lamé curves facilitate the computation of D(a) and H(a)?

Key findings

• A precise formula is derived relating D(a) and H(a) via a positive proportionality constant c_p, showing H(a) = c_p D(a) for all trivial critical points a.
• For half-period points (1/2 ω_i, 1/2 ω_j), the Hessian determinant is given by det D^2 G_2(p) = (4/(2π)^4) × |2e_i e_j + e_k^2 − 3e_k η_1|^2 + (2π/b) Re(3 ē_k (2e_i e_j + e_k^2 − 3e_k η_1)) with b = Im τ.
• At Weierstrass points (q, -q) with ℘(q) = ±√(g_2/12), the Hessian determinant is det D^2 G_2(p) = (9/π^4) |℘(q)|^2 (|℘(q) + η_1|^2 − (2π/b) Re(℘(q) + η_1)).
• The constant c_p is explicitly computed as c_p = e^{-c} |e_i - e_j| |e_i - e_k| |e_j - e_k| / (4bπ^4) > 0 for half-period points.
• For Weierstrass points, c_p = (9e^{-c}/(4bπ^4)) |℘(q)|^2 |℘'(q)|^4 ≥ 0, with c_p > 0 unless ℘(q) = 0, which occurs only when τ ≡ e^{πi/3}.
• The results confirm that both D(a) and H(a) are non-vanishing generically, implying local uniqueness of bubbling solutions under these conditions.