We develop Green's function estimate for manifolds satisfying a weighted Poincare inequality together with a compatible lower bound on the Ricci curvature. The estimate is then applied to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds. As an application, a Liouville property for finite energy holomorphic functions is proven on a class of complete Kahler manifolds. Consequently, such Kahler manifolds must be connected at infinity.

中文翻译：

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加权庞加莱不等式和泊松方程

我们为满足加权庞加莱不等式以及 Ricci 曲率的兼容下界的流形开发格林函数估计。然后应用该估计来建立此类流形上泊松方程解的存在性和精确估计。作为一个应用，有限能量全纯函数的刘维尔性质在一类完全 Kahler 流形上得到证明。因此，这种 Kahler 流形必须在无穷远处连接。