On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters

Abstract

We study the $SU(3)$ Toda system with singular sources \[ \begin{cases} Δu+2e^{u}-e^v=4π\sum_{k=0}^m n_{1,k}δ_{p_k}\quad\text{ on }\; E_τ,\\ Δv+2e^{v}-e^u=4π\sum_{k=0}^m n_{2,k}δ_{p_k}\quad\text{ on }\; E_τ, \end{cases} \] where $E_{\tau }:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$ with $\mathrm{Im}\tau >0$ is a flat torus, ${\delta }_{p_k}$ is the Dirac measure at $p_k$, and $n_{i,k}\in {\mathbb{Z}}_{\ge 0}$ satisfy ${\sum }_k{n}_{1,k}\not\equiv {\sum }_k{n}_{2,k}\mathrm{mod}3$. This is known as the non-critical case and it follows from a general existence result of \cite{BJMR} that solutions always exist. In this paper we prove that (i) The system has at most \[\frac{1}{3\times 2^{m+1}}\prod_{k=0}^m(n_{1,k}+1)(n_{2,k}+1)(n_{1,k}+n_{2,k}+2)\in\mathbb{N}\] solutions. We have several examples to indicate that this upper bound should be sharp. Our proof presents a nice combination of the apriori estimates from analysis and the classical Bézout theorem from algebraic geometry. (ii) For $m=0$ and $p_0=0$, the system has even solutions if and only if at least one of $\{n_{1,0},n_{2,0}\}$ is even. Furthermore, if $n_{1,0}$ is odd, $n_{2,0}$ is even and $n_{1,0}<n_{2,0}$, then except for finitely many $\tau$'s modulo $SL(2,\mathbb{Z})$ action, the system has exactly $\frac{n_{1,0}+1}{2}$ even solutions. Differently from \cite{BJMR}, our proofs are based on the integrability of the Toda system, and also imply a general non-existence result for even solutions of the Toda system with four singular sources.

Final version accepted for publication in JDG

References in corpus (1)

• Existence and multiplicity result for the singular Toda system