Rigidity results on Liouville equation

TL;DR

This paper classifies all bounded solutions of the Liouville equation in the plane, revealing discrete possible growth rates and establishing several symmetry and rigidity properties, including conditions for radiality and one-dimensionality.

Contribution

It provides a complete classification of bounded solutions and introduces new rigidity results, extending previous concavity rigidity to higher dimensions.

Findings

Solutions have discrete growth rate values: -2 or non-negative even integers.

Radial symmetry characterized by solutions tending to -∞ at infinity.

Concave, bounded solutions are necessarily one-dimensional.

Abstract

We give a complete classification of solutions bounded from above of the Liouville equation $$-\Delta u=e^{2u}\quad\mbox{in}\quad {\mathbf{R}}^2.$$ More generally, solutions in the class $$N:=\{ u:\limsup_{z\to\infty} u(z)/\log|z|:=k(u)<\infty\}$$ are described. As a consequence, we obtain five rigidity results. First, $k(u)$ can take only a discrete set of values: either $k=-2$, or $2k$ is a non-negative integer. Second, $u\to-\infty$ as $z\to\infty$, if and only if $u$ is radial about some point. Third, if $u$ is symmetric with respect to $x$ and $y$ axes and $u_x<0,\; u_y<0$ in the first quadrant then $u$ is radially symmetric. Fourth, if $u$ is concave and bounded from above, then $u$ is one-dimensional. Fifth, if $u$ is bounded from above, and the diameter of ${\mathbf{R}}^2$ with the metric $e^{2u}\delta$ is $\pi$, where $\delta$ is the Euclidean metric, then $u$ is either radial about a point or one-dimensional. In addition, we extend the concavity rigidity result on Liouville equation in higher dimensions.