An area growth estimate of the Liouville equation

We establish an area growth estimate for solutions that are bounded from above of the Liouville equation $\mathrm{\Delta }u+K{e}^{2u}=0$ with a positive pinched curvature $0<\lambda \le K\le \mathrm{\Lambda }$. 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.