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arXiv · 2409.19327

An area growth estimate of the Liouville equation

Xiaohan Cai·Mijia Lai·Chilin Zhang
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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.

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Xiaohan Cai, Mijia Lai, Chilin Zhang. 2024-09-28. An area growth estimate of the Liouville equation. https://doi.org/10.4310/mrl.260615144842

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