On conformal metrics of constant positive curvature in the plane

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.

18 pages, 2 figures

Cited by in corpus (1)

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