  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.
