On conformal metrics of constant positive curvature in the plane

We prove three theorems about solutions of $\mathrm{\Delta }u+e^{2u}=0$ 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 $e^u\mathrm{\mid }dz\mathrm{\mid }$ is at least $4\pi \mathrm{/}3$, except for two explicitly described families of solutions u.