In this paper, we study the properties of the first global term in the polyhomogeneous expansions for Liouville's equation. We obtain rigidity and gap results for the boundary integral of the global coefficient. We prove that such a boundary integral is always nonpositive, and is zero if and only if the underlying domain is a disc. More generally, we prove some gap theorems relating such a boundary integral to the number of components of the boundary. The conformal structure plays an essential role. We also give some positive mass theorem type results through the integral of the global coefficient.

中文翻译：

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刘维尔方程的刚性和间隙定理

在本文中，我们研究了刘维尔方程的多齐次展开中第一个全局项的性质。我们获得了全局系数边界积分的刚性和间隙结果。我们证明这样的边界积分总是非正的，并且当且仅当基础域是圆盘时才为零。更一般地说，我们证明了一些间隙定理，将这种边界积分与边界的分量数量相关联。共形结​​构起着至关重要的作用。我们还通过全局系数的积分给出了一些正质量定理类型的结果。