论文标题
2x2非对称单数liouville系统的学位计算定理
Degree Counting Theorems for 2x2 non-symmetric singular Liouville Systems
论文作者
论文摘要
令$（m，g）$是一个没有边界的紧凑riemann表面，$ u =（u_1，u_2）$是以下单一单数liouville系统的解决方案：$$Δ_gu_i+sum_+sum_+\ sum_ {j = 1}^2 A__ {IJ} h_je^{u_j}dV_g}-1)=\sum_{l=1}^{N}4πγ_l(δ_{p_l}-1), $$ where $h_1,h_2$ are positive smooth functions, $p_1,\cdots,p_N$ are distinct points on $M$, $δ_{p_l}$ are Dirac masses, $ρ=（ρ_1，ρ_2）（ρ_i\ geq 0）$和$（γ_1，\ cdots，γ_n）（γ_l> -1）$是常数向量。在以前的工作中，当$ a $满足标准假设时，我们为单数liouville系统提供了学位计数公式。在本文中，当系数矩阵$ a $不对称且不可固化时，我们建立了2 $ \ times $ 2的单个liouville系统的更一般程度计数公式。最后，解决方案的存在可以通过学位计数公式证明，该公式仅取决于域的拓扑和$ρ$的位置。
Let $(M,g)$ be a compact Riemann surface with no boundary and $u=(u_1,u_2)$ be a solution of the following singular Liouville system: $$Δ_g u_i+\sum_{j=1}^2 a_{ij}ρ_j(\frac{h_je^{u_j}}{\int_M h_je^{u_j}dV_g}-1)=\sum_{l=1}^{N}4πγ_l(δ_{p_l}-1), $$ where $h_1,h_2$ are positive smooth functions, $p_1,\cdots,p_N$ are distinct points on $M$, $δ_{p_l}$ are Dirac masses, $ρ=(ρ_1,ρ_2)(ρ_i\geq 0)$ and $(γ_1,\cdots,γ_N)(γ_l > -1)$ are constant vectors. In the previous work, we derive a degree counting formula for the singular Liouville system when $A$ satisfies standard assumptions. In this article, we establish a more general degree counting formula for 2$\times$2 singular Liouville system when the coefficient matrix $A$ is non-symmetric and non-invertible. Finally, the existence of solution can be proved by the degree counting formula which depends only on the topology of the domain and the location of $ρ$.
