On conformal metrics of constant positive curvature in the plane

TL;DR

This paper characterizes solutions to a specific nonlinear PDE related to conformal metrics of constant positive curvature in the plane, providing explicit classifications and geometric bounds.

Contribution

It explicitly classifies all concave and quasiconcave solutions and establishes a lower bound on the plane's diameter under these metrics, with detailed solution families.

Findings

Explicit classification of concave solutions

Explicit classification of quasiconcave solutions

Lower bound of 4π/3 on the plane's diameter under the metric

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.