Some geometric inequalities related to Liouville equation

Abstract

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 ${\int }_{{\mathbb{R}}^2}{e}^{2u}dx<\infty$, 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.

This article was written in 2020 and still under review. We forgot to post it to arXiv. Later several experts learned about our results and have inquired us about the paper, so we now post it here