On conformal metrics of constant positive curvature in the plane

We prove three theorems about solutions of ∆ u + e 2 u = 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 π/ 3, except for two explicitly described families of solutions u . 2010 MSC 35B99

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