An area growth estimate of the Liouville equation

TL;DR

This paper derives an area growth estimate for bounded solutions to the Liouville equation with positive pinched curvature, providing new proofs and classifications of solutions in specific geometric contexts.

Contribution

It introduces a novel area growth estimate for solutions of the Liouville equation under curvature bounds and applies it to classify solutions in the half-plane.

Findings

Established an area growth estimate for bounded solutions.

Provided a new proof of a known classification result.

Classified solutions with boundary constant geodesic curvature.

Abstract

We establish an area growth estimate for solutions that are bounded from above of the Liouville equation $\Delta u+K e^{2u}=0$ with a positive pinched curvature $0<\lambda\leq K\leq\Lambda$. As an application, we provide a new proof of Eremenko-Gui-Li-Xu's result in [EGLX]. We also classify solutions with an upper bound in the half plane with the boundary having constant geodesic curvature.