Blow--up Solutions of Liouville's Equation and Quasi--Normality

TL;DR

This paper establishes that families of meromorphic functions with bounded spherical area are quasi-normal, connecting this to blow-up solutions of Liouville's equation and Gromov's compactness theorem.

Contribution

It proves the quasi-normality of families of meromorphic functions with bounded spherical area and relates this to blow-up solutions of Liouville's equation.

Findings

Family $_C(D)$ is quasi-normal of order ≤ C.

Connections made to blow-up solutions of Liouville's equation.

Relation to Gromov's compactness theorem.

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 Br\'ezis 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.