Abstract
We prove that the family \(\mathcal {F}_C(D)\) of all meromorphic functions f on a domain \(D\subseteq \mathbb {C}\) with the property that the spherical area of the image domain f(D) is uniformly bounded by \(C \pi \) is quasi-normal of order \(\le C\). We also discuss the close relations between this result and the well-known work of Brézis and Merle on blow-up solutions of Liouville’s equation. These results are completely in the spirit of Gromov’s compactness theorem, as pointed out at the end of the paper.
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