Exact number and non-degeneracy of critical points of multiple Green functions on rectangular tori
type: Item
instance of: scholarly article
title: Exact number and non-degeneracy of critical points of multiple Green functions on rectangular tori
zbMATH Open document ID: 1482.35081
author: Zhijie Chen
author: Chang-Shou Lin
published in: Journal of Differential Geometry
publication date: 2021-09-08T00:00:00Z
review text: The authors investigate the Green function on a flat torus. Specifically, let \(E_{\tau} := \mathbb{C}/\Lambda_{\tau}\) be a torus in the plane, where \(\Lambda_{\tau} = \mathbb{Z}\omega_1 + \mathbb{Z}\omega_2\) with \(\omega_1 = 1\), \(\omega_2 = \tau \) and \(\tau \in \mathbb{H} = \{\tau \; | \; \mathrm{Im} \tau > 0\}\). Let \(G(z) = G(z; \tau)\) be the Green function on \(E_{\tau}\) defined by \[ -\Delta G(z;\tau)= \delta_0 - \frac{1}{|E_{\tau}|}\quad\mbox{on}\;\; E_{\tau}\,,\quad \int_{E_{\tau}} G(z; \tau) = 0, \] where \(\delta_0\) is the Dirac measure at \(0\) and \(|E_{\tau}|\) is the area of the torus \(E_{\tau}\). Let \(E_\tau^\times := E_{\tau}\setminus\{0\}\) and consider the complete diagonal in \((E_\tau^\times)^n \): \[ \Delta_n := \{(z_1, \cdots, z_n)\in(E_\tau^\times)^n \,|\, z_i = z_j \;\;\mbox{ for some} \;\; i \not= j\}. \] Let \(\mathbf{z}=(z_1,\cdots,z_n)\in (E_\tau^\times)^n\setminus \Delta_n\) and define the multiple Green function \(G_n(\mathbf{z}) = G_n(\mathbf{z}; \tau )\) on \((E_\tau^\times)^n\setminus \Delta_n\) by \[ G_n(\mathbf{z}; \tau ) :=\sum_{i<j} G(z_i -z_j ; \tau ) - n \sum^n_{i=1} G(z_i; \tau ).\eqno{(1.1)} \] The main result is as follows. Theorem 1.1. Let \(\tau \in i\mathbb{R}_{>0}\). Then the followings hold. \begin{itemize} \item[(a)] \(G_n(\mathbf{z}; \tau )\) has exactly \(2n+1\) critical points \(\mathbf a\), which are all trivial critical points and satisfy \(\mathbf{a} = \overline{\mathbf{a}}\). \item[(b)] The Hessian \(\det D^2G_n(\mathbf{a}; \tau ) \not= 0\) for any critical point \(\mathbf{a}\). \item[(c)] \(D(\mathbf{a}; \tau ) \not= 0\) for any critical point \(\mathbf{a}\). \end{itemize} Here writing \(\mathbf{a}=\{a_1,\cdots ,a_n\}\), \(\mathbf{a} = \overline{\mathbf{a}}\) means \[ \{ a_1, \cdots, a_n\} = \{\overline{a}_1,\cdots ,\overline{a}_n\}\quad \mbox{in}\quad E_{\tau}. \] The proof is based on the connection between \(G_n\) and the classical Lamé equation, and one key step is to establish a precise formul…
reviewed by: Petr Tomiczek
Mathematics Subject Classification ID: 35J08
zbMATH DE Number: 7392772
zbMATH Keywords: Green function
zbMATH Keywords: critical points
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1625860623
OpenAlex ID: W3181915746
Wikidata QID: Q115164271
cites work: Uniqueness of bubbling solutions of mean field equations
cites work: Vortex condensation in the Chern-Simons Higgs model: An existence theorem
cites work: A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description
cites work: Mean field equations, hyperelliptic curves and modular forms. I
cites work: Sharp estimates for solutions of multi‐bubbles in compact Riemann surfaces
cites work: Topological degree for a mean field equation on Riemann surfaces
cites work: Mean Field Equation of Liouville Type with Singular Data: Topological Degree
cites work: Q5489520
cites work: The geometry of generalized Lamé equation. I
cites work: Green function, Painlevé VI equation, and Eisenstein series of weight one
cites work: Sharp Nonexistence Results for Curvature Equations with Four Singular Sources on Rectangular Tori
cites work: Picard potential and Hill's equation on a torus
cites work: Lamé Potentials and the Stationary (m)KdV Hierarchy
cites work: Treibich-Verdier potentials and the stationary (m)KdV hierarchy
cites work: VII—Further Investigations into the Periodic Lamé Functions
cites work: Elliptic functions, Green functions and the mean field equations on tori
cites work: Mean field equations, hyperelliptic curves and modular forms. II
cites work: Geometric quantities arising from bubbling analysis of mean field equations
cites work: On the Minimality of Extra Critical Points of Green Functions on Flat Tori
cites work: Existence of bubbling solutions for Chern-Simons model on a torus
cites work: Double vortex condensates in the Chern-Simons-Higgs theory
cites work: Elementary Dirichlet Series and Modular Forms
cites work: The Heun equation and the Calogero-Moser-Sutherland system. I: The Bethe ansatz method
cites work: The Heun Equation and the Calogero-Moser-Sutherland System III: The Finite-Gap Property and the Monodromy
cites work: The Heun equation and the Calogero-Moser-Sutherland system. IV: The Hermite-Krichever ansatz
cites work: The Heun equation and the Calogero-Moser-Sutherland system V: generalized Darboux transformations
cites work: Exceptional coverings and sums of 4 triangular numbers
DOI: 10.4310/JDG/1625860623
description: scientific article; zbMATH DE number 7392772