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Blow-Up Solutions of Liouville’s Equation and Quasi-Normality

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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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Correspondence to Oliver Roth.

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Communicated by Elias Wegert.

Dedicated to the Memory of Professor Stephan Ruscheweyh—our Teacher, Mentor, and Friend

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Grahl, J., Kraus, D. & Roth, O. Blow-Up Solutions of Liouville’s Equation and Quasi-Normality. Comput. Methods Funct. Theory 20, 677–692 (2020). https://doi.org/10.1007/s40315-020-00339-4

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