Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Green functions, Hitchin's formula and curvature equations on tori

Zhijie Chen, Erjuan Fu, Chang-Shou Lin

TL;DR

This work investigates the Green function $G(z;\tau)$ on flat tori and its averaged counterpart $G_p(z)=\tfrac12(G(z-p)+G(z+p))$, linking critical-point structures to curvature equations on $E_\tau$. Building on Lin–Wang’s $3$ or $5$ point dichotomy and Bergweiler–Eremenko’s anti-holomorphic dynamics, it proves that the $5$-point case is generically nondegenerate and that $G_p$ can have $4$, $6$, $8$, or $10$ critical points, with a refined bound to $\{4,6\}$ under Hitchin-type relations. Generic non-degeneracy of critical points is established via Hitchin’s formula, Fatou theory, and Sard’s/deegree arguments, enabling a detailed description of how critical-point counts vary with the torus shape and the parameter $p$. The paper then connects these classical aspects to the curvature mean-field equation $\Delta u+e^{u}=4\pi(\delta_p+\delta_{-p})$ on $E_\tau$, proving a one-to-one correspondence between pairs of nontrivial critical points of $G_p$ and even one-parameter families of solutions, thereby clarifying bubbling behavior, solution multiplicity, and monodromy (unitary) aspects through Painlevé VI phenomena.

Abstract

Let $G(z)=G(z;τ)$ be the Green function on the flat torus $E_τ=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ with the singularity at $0$. Lin and Wang (Ann. Math. 2010) proved that $G(z)$ has either $3$ or $5$ critical points (depending on the choice of $τ$). Later, Bergweiler and Eremenko (Proc. Amer. Math. Soc. 2016) gave a new proof of this remarkable result by using anti-holomorphic dynamics. In this paper, firstly, we prove that once $G(z)$ has $5$ critical points, then these $5$ critical points are all non-degenerate. Secondly, we study the sum of two Green functions which can be reduced to $G_p(z):=\frac12(G(z+p)+G(z-p))$. We prove that for any $p$ satisfying $p\neq -p$ in $E_τ$, the number of critical points of $G_p(z)$ belongs to $\{4,6,8,10\}$ (depending on the choice of $(τ, p)$) and each number really occurs. We apply Hitchin's formula (J. Differ. Geom. 1995) in a surprising way to prove the generic non-degeneracy of critical points. This allows us to study the distribution of the numbers of critical points of $G_p(z)$ as $p$ varies. Applications to the curvature equation $Δu+e^{u}=4π(δ_{p}+δ_{-p})$ on $E_τ$ are also given, and how the geometry of the torus affects the solution structure is studied.

Green functions, Hitchin's formula and curvature equations on tori

TL;DR

This work investigates the Green function on flat tori and its averaged counterpart , linking critical-point structures to curvature equations on . Building on Lin–Wang’s or point dichotomy and Bergweiler–Eremenko’s anti-holomorphic dynamics, it proves that the -point case is generically nondegenerate and that can have , , , or critical points, with a refined bound to under Hitchin-type relations. Generic non-degeneracy of critical points is established via Hitchin’s formula, Fatou theory, and Sard’s/deegree arguments, enabling a detailed description of how critical-point counts vary with the torus shape and the parameter . The paper then connects these classical aspects to the curvature mean-field equation on , proving a one-to-one correspondence between pairs of nontrivial critical points of and even one-parameter families of solutions, thereby clarifying bubbling behavior, solution multiplicity, and monodromy (unitary) aspects through Painlevé VI phenomena.

Abstract

Let be the Green function on the flat torus with the singularity at . Lin and Wang (Ann. Math. 2010) proved that has either or critical points (depending on the choice of ). Later, Bergweiler and Eremenko (Proc. Amer. Math. Soc. 2016) gave a new proof of this remarkable result by using anti-holomorphic dynamics. In this paper, firstly, we prove that once has critical points, then these critical points are all non-degenerate. Secondly, we study the sum of two Green functions which can be reduced to . We prove that for any satisfying in , the number of critical points of belongs to (depending on the choice of ) and each number really occurs. We apply Hitchin's formula (J. Differ. Geom. 1995) in a surprising way to prove the generic non-degeneracy of critical points. This allows us to study the distribution of the numbers of critical points of as varies. Applications to the curvature equation on are also given, and how the geometry of the torus affects the solution structure is studied.

Paper Structure

This paper contains 14 sections, 38 theorems, 200 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Green function $G$
  3. The Green function $G_p$
  4. The curvature equation
  5. Possible numbers of critical points
  6. Criterion for degeneracy
  7. Criterion for degeneracy of trivial critical points
  8. Examples that evey number in $\{4,6,8,10\}$ occurs
  9. Generic non-degeneracy of critical points
  10. Counting the number of critical points
  11. Proofs of Theorems \ref{['thm-section5']}, \ref{['thm-section5-5']} and Theorem \ref{['main-thm-2']} (3-1)-(3-2)
  12. Structure of critical points for $|\wp(p)|$ large
  13. Two special examples
  14. Connection with the curvature equation

Key Result

Theorem 1.1

Let $G(z)=G(z;\tau)$ be the Green function of the torus $E_\tau$ and let $p\in E_{\tau}\setminus E_{\tau}[2]$.

Figures (2)

  • Figure 1: The four disks and $\{e_1,e_2,e_3\}$ for $\tau=e^{\pi i/3}=\frac{1}{2}+\frac{\sqrt{3}}{2}i$: the smallest circle for $\partial\mathcal{B}_0$, left for $\partial\mathcal{B}_1$, upper right for $\partial\mathcal{B}_2$ and lower right for $\partial\mathcal{B}_3$.
  • Figure 2: The four disks and $\{e_1,e_2,e_3\}$ for $\tau=i$: the smallest circle for $\partial\mathcal{B}_0$, biggest for $\partial\mathcal{B}_3$, left for $\partial\mathcal{B}_1$ and right for $\partial\mathcal{B}_2$.

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem A
  • Theorem B
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • proof
  • ...and 68 more