具有四个奇点的环面上的球面度量分类，I：半周期
Classification of spherical metrics on tori with four singularities, I: half periods
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中文总结 AI 辅助
研究环面\(E_\tau\)上特定球面度量分类问题，通过构造多重格林函数\(G_2\)，将相关方程解分为特殊和非特殊两类，得出特殊解性质及非特殊解存在条件，还确定了\(G_2\)临界点及退化准则。
中文摘要 AI 辅助
对环面\(E_\tau\)上在每个半周期点\(\frac{\omega_k}{2}, k = 0,1,2,3\)处具有\(4\pi\)圆锥角的球面度量进行分类，等同于对以下曲率方程的解进行分类：\(\Delta u + e^u = 4\pi\sum_{k = 0}^3\delta_{\frac{\omega_k}{2}}\) 在\(E_\tau\)上，其中\(\tau\in \mathbb{H}:=\{z\in \mathbb{C}\mid \mathrm{Im} \, z>0\}\)且\(\delta_p\)是\(p\in E_\tau\)处的狄拉克测度。通过用\(E_\tau\)上的格林函数\(G(z;\tau)\)构造多重格林函数\(G_2(z_1, z_2;\tau)\)，将方程的解分为特殊和非特殊两类。特殊解是偶函数且特殊解集合与\(SL(2,\mathbb{C})/SU(2)\)同构；非特殊解当且仅当\(\tau\in \mathcal{E}\)时存在，且有六个单参数族。还确定了\(G_2\)的临界点及退化准则。
英文摘要
Classifying the spherical metrics on a torus $E_τ$ with $4π$ conic angle at each half period point\, ${ω_k}/{2}, k=0,1,2,3$\, is equivalent to classify solutions of the following curvature equation \begin{align}\label{eq0731093154} Δu+e^u=4π\sum_{k=0}^3δ_{\frac{ω_k}{2}}\text{\ on\ }E_τ\end{align} where $τ\in \mathbb{H}:=\{z\in \mathbb{C}\mid \mathrm{Im} \, z>0\}$ and $δ_p$ is the Dirac measure at $p\in E_τ$. By constructing a multiple Green function $$G_2(z_1, z_2;τ):=G(z_1-z_2;τ)-\frac{1}{2}\sum_{j=0}^3\left(G(z_1-\frac{ω_j}{2};τ)+G(z_2-\frac{ω_j}{2};τ)\right), $$ in terms of the Green function $G(z;τ)$ on $E_τ$, 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 $τ\in \mathbb{H}$. \item a non-special solution exists if and only if $τ\in \mathcal{E}$. Moreover, if $τ\in \mathcal{E}$, then there are six one-parameter families of nonspecial solutions. \end{enumerate} where $$\mathcal{E}:=\left\{τ\in \mathbb{H}\mid G(z;τ) \,\,\text{has exactly 5 critical points.}\right\}.$$ 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.
