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arXiv · 2208.04672

On conformal metrics of constant positive curvature in the plane

Walter Bergweiler·Alexandre Eremenko·James Langley
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Abstract

We prove three theorems about solutions of $\Delta u + e^{2u} = 0$ in the plane. The first two describe explicitly all concave and quasiconcave solutions. The third theorem says that the diameter of the plane with respect to the metric with line element $e^{u}|dz|$ is at least $4\pi/3$, except for two explicitly described families of solutions u.

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Walter Bergweiler, Alexandre Eremenko, James Langley. 2022-08-09. On conformal metrics of constant positive curvature in the plane. https://doi.org/10.15407/mag19.01.059

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