Some geometric inequalities related to Liouville equation

TL;DR

This paper investigates geometric inequalities related to solutions of the Liouville equation in two dimensions, establishing bounds on the conformal diameter of the plane and constructing solutions that attain these bounds, with extensions to supersolutions and higher dimensions.

Contribution

It provides explicit bounds on the conformal diameter for solutions and supersolutions of the Liouville equation, including construction of extremal solutions and connections to geometric inequalities.

Findings

Diameter of b5 under conformal metric b5 is at least b5 for solutions.

Constructed solutions with diameters ranging from b5 to 2b5.

Supersolutions with finite integral have diameter at most 2b5.

Abstract

In this paper, we prove that if $u$ is a solution to the Liouville equation \begin{align} \label{scalliouville} \Delta u+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.