Classification of spherical metrics on tori with four singularities, I: half periods

Abstract

Classifying the spherical metrics on a torus $E_\tau$ with $4\pi$ conic angle at each half period point\, ${\omega_k}/{2}, k=0,1,2,3$\, is equivalent to classify solutions of the following curvature equation \begin{align}\label{eq0731093154} \Delta u+e^u=4\pi\sum_{k=0}^3\delta_{\frac{\omega_k}{2}}\text{\ on\ }E_\tau \end{align} where $\tau\in \mathbb{H}:=\{z\in \mathbb{C}\mid \mathrm{Im} \, z>0\}$ and $\delta_p$ is the Dirac measure at $p\in E_\tau$. By constructing a multiple Green function $$G_2(z_1, z_2;\tau):=G(z_1-z_2;\tau)-\frac{1}{2}\sum_{j=0}^3\left(G(z_1-\frac{\omega_j}{2};\tau)+G(z_2-\frac{\omega_j}{2};\tau)\right), $$ in terms of the Green function $G(z;\tau)$ on $E_\tau$, we classify the solutions of (\ref{eq0731093154}) into two types: \emph{special} and \emph{non-special}. Furthermore, we obtain the following conclusion about the solutions of (\ref{eq0731093154}): \begin{enumerate} \item any special solution is an even function and the set of special solutions is isomorphic to $SL(2,\mathbb{C})/SU(2)$ for all $\tau\in \mathbb{H}$. \item a non-special solution exists if and only if $\tau\in \mathcal{E}$. Moreover, if $\tau\in \mathcal{E}$, then there are six one-parameter families of nonspecial solutions. \end{enumerate} where $$\mathcal{E}:=\left\{\tau\in \mathbb{H}\mid G(z;\tau) \,\,\text{has exactly 5 critical points.}\right\}.$$ The set $\mathcal{E}$ is completely determined in \cite{CLW2018, Lin}, which is a union of countable many open triangular domains. As a byproduct, we completely determine and classify the critical points of $G_2$ and then obtain the degeneracy criterion of critical points for $G_2$, which may be of independent interest.