平环上多重格林函数临界点的结构及其应用
Structure of critical points of multiple Green functions on flat torus and applications
浏览论文内容
中文总结 AI 辅助
本文证明平环上多重格林函数至多有n对非平凡临界点且上界最优，并应用于曲率方程与积分Lamé方程，揭示线性化算子核一维性。
中文摘要 AI 辅助
设 $E_{\tau}:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ 为平环，$G(z)=G(z;\tau)$ 为 $E_{\tau}$ 上以 $0$ 为奇点的格林函数。考虑 $(E_{\tau})^{n}$ 上的多重格林函数 $G_{n}$：\\[ G_{n}(z_{1},\cdots,z_{n}):=\sum_{i<j}G(z_{i}-z_{j})-n\sum_{i=1}^{n}G(z_{i})。\\] 若在 $E_{\tau}$ 中满足 $\{z_{1},\cdots,z_{n}\}=\{-z_{1},\cdots,-z_{n}\}$，则称临界点为平凡的。Lin 和 Wang（Ann. Math. 2010）证明了 $G_1=-G$ 恰好有三个平凡临界点，并且至多有一对非平凡临界点（取决于 $\tau$ 的选择）。对于一般 $n\geq 2$，Chai、Lin 和 Wang（Camb. J. Math. 2015）证明了 $G_n$ 至多有 $2n+1$ 个平凡临界点，但非平凡临界点可能有多少个在当时完全未解决。在本文中，我们证明 $G_n$ 至多有 $n$ 对非平凡临界点，且此上界至少在 $n=2,3$ 时是最优的。关键思想是证明非平凡临界点（若存在）总是非退化的，并且贡献相同的度数 $(-1)^{n+1}$。我们将给出对曲率方程 $\Delta u+e^{u}=8\pi n \delta_{0}$ 和积分 Lamé 方程 $y''=[n(n+1)\wp(z)+B]y$ 的应用。特别地，我们证明对于曲率方程的任何解，线性化算子的核是一维的。
英文摘要
Let $E_τ:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ be a flat torus and $G(z)=G(z;τ)$ be the Green function on $E_τ$ with the singularity at $0$. Consider the multiple Green function $G_{n}$ on $(E_τ)^{n}$: \[ G_{n}(z_{1},\cdots,z_{n}):=\sum_{i<j}G(z_{i}-z_{j})-n\sum_{i=1}^{n}G(z_{i}). \] A critical point is called trivial if $\{z_{1},\cdots,z_{n}\}=\{-z_{1},\cdots,-z_{n}\}$ in $E_τ$. Lin and Wang (Ann. Math. 2010) proved that $G_1=-G$ has exactly three trivial critical points and at most one pair of nontrivial critical points (depends on the choice of $τ$). For general $n\geq 2$, Chai, Lin ang Wang (Camb. J. Math. 2015) proved that $G_n$ has at most $2n+1$ trivial critical points, but how many nontrivial critical points might exist remained completely open there. In this paper, we prove that $G_n$ has at most $n$ pairs of nontrivial critical points, and this upper bound is optimal at least for $n=2,3$. The key idea is to show that nontrivial critical points (if exist) are always non-degenerate and contribute the same degree $(-1)^{n+1}$. Applications to the curvature equation $Δu+e^{u}=8πn δ_{0}$ and the integral Lamé equation $y''=[n(n+1)\wp(z)+B]y$ will be given. In particular, we show that the kernel of the linearized operator is one-dimensional for any solution of the curvature equation.
发表机构
- Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
- National Taiwan University（台湾大学）
机构由 AI 辅助整理，请以论文原文为准。
