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

TL;DR

This paper investigates the critical points of Green functions on tori, proving non-degeneracy in certain cases, analyzing sums of Green functions, and applying Hitchin's formula to understand their distribution and related curvature equations.

Contribution

It establishes the non-degeneracy of five critical points of Green functions and analyzes the critical point count of their sums using Hitchin's formula, connecting to curvature equations on tori.

Findings

Five critical points are non-degenerate when they exist.

Number of critical points of combined Green functions varies between 4 and 10.

Application of Hitchin's formula reveals distribution patterns of critical points.

Abstract

Let $G(z)=G(z;\tau)$ be the Green function on the flat torus $E_{\tau}=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ 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 $\tau$). 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 $E_{\tau}$, the number of critical points of $G_p(z)$ belongs to $\{4,6,8,10\}$ (depending on the choice of $(\tau, p)$) 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 $\Delta u+e^{u}=4\pi(\delta_{p}+\delta_{-p})$ on $E_{\tau}$ are also given, and how the geometry of the torus affects the solution structure is studied.