Exact number and non-degeneracy of critical points of multiple Green functions on rectangular tori
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 formula of the Hessian of critical points of \(G_n\) in terms of the monodromy data of the Lamé equation. The proof also determines the exact number of critical points satisfying \(D(\mathbf{a}; \tau ) < 0\). As an application, the authors show that the mean field equation \[ \Delta u+e^u=\varrho\delta_0\quad\mbox{on}\;\;E_{\tau} \] has exactly \(n\) solutions for \(8\pi n - \varrho > 0\) small, and exactly \(n + 1\) solutions for \(\varrho -8\pi n > 0\) small.
scientific article; zbMATH DE number 7392772
scientific article