An area growth estimate of the Liouville equation
X.D. Cai, Mijia Lai, Chilin Zhang · Mathematical Research Letters · 2026
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 classification result in [7]. We also classify solutions with an upper bound in the half plane with the boundary having constant geodesic curvature.