Some geometric inequalities related to Liouville equation

In this paper, we prove that if $u$ is a solution to the Liouville equation \begin{align} \label{scalliouville} Δu+e^{2u} =0 \quad \mbox{in ${\mathbb{R}}^2$,} \end{align}then the diameter of ${\mathbb{R}}^2$ under the conformal metric $g=e^{2u}\delta$ is bounded below by $\pi$. Here $\delta$ is the Euclidean metric in ${\mathbb{R}}^2$. Moreover, we explicitly construct a family of solutions such that the corresponding diameters of ${\mathbb{R}}^2$ range over $[\pi ,2\pi )$. We also discuss supersolutions. We show that if $u$ is a supersolution and ${\int }_{{\mathbb{R}}^2}{e}^{2u}dx<\mathrm{\infty }$, then the diameter of ${\mathbb{R}}^2$ under the metric $e^{2u}\delta$ is less than or equal to $2\pi$. For radial supersolutions, we use both analytical and geometric approaches to prove some inequalities involving conformal lengths and areas of disks in ${\mathbb{R}}^2$. We also discuss the connection of the above results with the sphere covering inequality in the case of Gaussian curvature bounded below by $1$. Higher dimensional generalizations are also discussed.