On Conformal Metrics of Constant Positive Curvature in the Plane

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. third theorem says that diameter plane with respect to metric line element $e^{u}|dz|$ is at least $4\pi/3$, except for described families u.